Explanation:
Okay, so I need to simplify this big fraction where the numerator is the product of terms like (2^4 + 1/4), (4^4 + 1/4), all the way up to (2n)^4 + 1/4, and the denominator is the product of similar terms but with odd numbers: (1^4 + 1/4), (3^4 + 1/4), ..., up to (2n - 1)^4 + 1/4.

First, let me write down the expression more clearly:

$$
\frac{(2^4 + \frac{1}{4})(4^4 + \frac{1}{4}) \dotsm [(2n)^4 + \frac{1}{4}]}{(1^4 + \frac{1}{4})(3^4 + \frac{1}{4}) \dotsm [(2n - 1)^4 + \frac{1}{4}]}
$$

So, the numerator is the product of (2k)^4 + 1/4 for k from 1 to n, and the denominator is the product of (2k - 1)^4 + 1/4 for k from 1 to n.

Hmm, maybe I can first try to simplify each term in the numerator and denominator. Let me take a general term, say, for a number x, x^4 + 1/4. Maybe there's a way to factor this expression?

I remember that x^4 + 4y^4 can be factored using the Sophie Germain identity: x^4 + 4y^4 = (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2). Wait, let me check that.

Yes, the Sophie Germain identity is x^4 + 4y^4 = (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2). So if I can write x^4 + 1/4 in terms of this identity, maybe?

Let me see. Let me set 4y^4 = 1/4. Then y^4 = 1/16, so y = 1/2. Therefore, x^4 + 1/4 = x^4 + 4*(1/2)^4. So that would be equal to (x^2 + 2x*(1/2) + 2*(1/2)^2)(x^2 - 2x*(1/2) + 2*(1/2)^2). Let me compute that.

First term: x^2 + 2x*(1/2) + 2*(1/2)^2 = x^2 + x + 2*(1/4) = x^2 + x + 1/2.

Second term: x^2 - 2x*(1/2) + 2*(1/2)^2 = x^2 - x + 1/2.

Therefore, x^4 + 1/4 factors into (x^2 + x + 1/2)(x^2 - x + 1/2).

So, for each term in the numerator and denominator, we can factor them like that. Let me check with a specific example. Let me take x = 2. Then 2^4 + 1/4 = 16 + 1/4 = 65/4. On the other hand, (2^2 + 2 + 1/2)(2^2 - 2 + 1/2) = (4 + 2 + 0.5)(4 - 2 + 0.5) = (6.5)(2.5) = 16.25, which is 65/4. Correct. So that works.

Similarly, for x = 1: 1^4 + 1/4 = 1 + 1/4 = 5/4. Factoring: (1 + 1 + 1/2)(1 - 1 + 1/2) = (2.5)(0.5) = 1.25 = 5/4. Correct.

So, this factorization works. Therefore, each term in the numerator and denominator can be written as a product of two quadratics.

Therefore, the entire expression can be written as:

Numerator: product_{k=1}^n [( (2k)^2 + 2k + 1/2 ) * ( (2k)^2 - 2k + 1/2 ) ]

Denominator: product_{k=1}^n [ ( (2k - 1)^2 + (2k - 1) + 1/2 ) * ( (2k - 1)^2 - (2k - 1) + 1/2 ) ]

So, if I can write each term in numerator and denominator as such, maybe there is some telescoping or cancellation when taking the ratio.

Let me compute the terms in the numerator and denominator separately. Let me first compute the factors for the numerator terms. Let me denote for each term in the numerator, which is (2k)^4 + 1/4, which factors into:

[( (2k)^2 + 2*(2k) + 1/2 ) * ( (2k)^2 - 2*(2k) + 1/2 ) ] ?

Wait, no. Wait, earlier we had x^4 + 1/4 = (x^2 + x + 1/2)(x^2 - x + 1/2). So for x = 2k, it would be:

( (2k)^2 + (2k) + 1/2 ) * ( (2k)^2 - (2k) + 1/2 )

Wait, but earlier when I did x^4 + 1/4, the factors were (x^2 + x + 1/2)(x^2 - x + 1/2). So yes, that's correct. So for x = 2k, the factors are:

( (2k)^2 + 2k + 1/2 ) and ( (2k)^2 - 2k + 1/2 )

Similarly, for the denominator terms, which are (2k - 1)^4 + 1/4, so x = 2k - 1, so factors are:

( (2k - 1)^2 + (2k - 1) + 1/2 ) * ( (2k - 1)^2 - (2k - 1) + 1/2 )

So, perhaps if I can write these factors in terms of something that telescopes when taking the product over k.

Alternatively, maybe I can write each quadratic term as something else. Let me compute them:

First, let me compute the first factor in the numerator term:

( (2k)^2 + 2k + 1/2 ) = 4k^2 + 2k + 1/2. Similarly, the second factor is 4k^2 - 2k + 1/2.

Similarly, for the denominator terms:

First factor: ( (2k - 1)^2 + (2k - 1) + 1/2 )

Let me compute that:

(2k - 1)^2 = 4k^2 - 4k + 1. Then adding (2k - 1) gives 4k^2 - 4k + 1 + 2k - 1 = 4k^2 - 2k. Then adding 1/2 gives 4k^2 - 2k + 1/2.

Similarly, the second factor for the denominator term is:

( (2k - 1)^2 - (2k - 1) + 1/2 )

Compute that:

(2k - 1)^2 - (2k - 1) + 1/2 = 4k^2 - 4k + 1 - 2k + 1 + 1/2 = Wait, let me do it step by step.

First, (2k - 1)^2 = 4k^2 - 4k + 1. Then subtract (2k - 1): 4k^2 - 4k + 1 - 2k + 1 = 4k^2 - 6k + 2. Then add 1/2: 4k^2 - 6k + 2 + 1/2 = 4k^2 - 6k + 5/2. Wait, that seems complicated. Wait, maybe I made a mistake here.

Wait, let me redo that. The second factor for the denominator term is:

( (2k - 1)^2 - (2k - 1) + 1/2 )

So:

First term: (2k - 1)^2 = 4k² - 4k + 1

Second term: - (2k - 1) = -2k + 1

Third term: + 1/2

So adding them all together:

4k² - 4k + 1 - 2k + 1 + 1/2 = 4k² - 6k + 2 + 1/2 = 4k² - 6k + 5/2.

Hmm, okay. So that's the second factor for the denominator term.

But maybe there's a better way to write these expressions. Let me check if these quadratic terms can be related to something else.

Alternatively, maybe I can write the numerator and denominator terms as products of terms that telescope when taking the ratio. Let me think.

Let me first look at the numerator factors. For each term in the numerator, we have two factors:

A_k = 4k² + 2k + 1/2 and B_k = 4k² - 2k + 1/2.

Similarly, for the denominator terms, each term is:

C_k = 4k² - 2k + 1/2 and D_k = 4k² - 6k + 5/2? Wait, no. Wait, earlier for the denominator terms, the first factor was 4k² - 2k + 1/2, which is same as B_k? Wait, no. Wait, for the denominator term, the first factor was:

( (2k - 1)^2 + (2k - 1) + 1/2 ) = 4k² - 2k + 1/2. Wait, earlier when I calculated that, I had:

(2k - 1)^2 + (2k - 1) + 1/2 = 4k² - 4k + 1 + 2k - 1 + 1/2 = 4k² - 2k + 1/2. Yes, so that is equal to 4k² - 2k + 1/2. Which is same as B_k? Wait, B_k was 4k² - 2k + 1/2. Yes! So the first factor in the denominator term is equal to B_k.

Wait, but the denominator term is (2k - 1)^4 + 1/4, which factors into ( (2k - 1)^2 + (2k - 1) + 1/2 ) * ( (2k - 1)^2 - (2k - 1) + 1/2 ) = C_k * D_k, where C_k = 4k² - 2k + 1/2 and D_k = 4k² - 6k + 5/2.

But the numerator term is (2k)^4 + 1/4 = A_k * B_k, where A_k = 4k² + 2k + 1/2 and B_k = 4k² - 2k + 1/2.

So, notice that the B_k in the numerator is same as C_k in the denominator. Therefore, when we take the ratio of numerator over denominator, each term in the numerator has a B_k, and each term in the denominator has a C_k. But since C_k is same as B_k, perhaps there is some cancellation? Wait, but the denominator has terms for k from 1 to n, and the numerator also has terms for k from 1 to n.

Wait, but let me think. Let me write the entire product:

Numerator: product_{k=1}^n [A_k * B_k] = product_{k=1}^n A_k * product_{k=1}^n B_k

Denominator: product_{k=1}^n [C_k * D_k] = product_{k=1}^n C_k * product_{k=1}^n D_k

But since C_k = B_k, then product_{k=1}^n C_k = product_{k=1}^n B_k. Therefore, the ratio becomes:

[ product_{k=1}^n A_k * product_{k=1}^n B_k ] / [ product_{k=1}^n B_k * product_{k=1}^n D_k ] = [ product_{k=1}^n A_k ] / [ product_{k=1}^n D_k ]

Therefore, the ratio simplifies to the product of A_k divided by the product of D_k.

So now, the problem reduces to computing:

Product_{k=1}^n [A_k / D_k] where A_k = 4k² + 2k + 1/2 and D_k = 4k² - 6k + 5/2.

Wait, but maybe I can write A_k and D_k in terms of something else. Let me check if there's a relation between A_k and D_{k} or D_{k} and A_{k+1} or something like that.

Let me compute A_k and D_k:

A_k = 4k² + 2k + 1/2

D_k = 4k² - 6k + 5/2

Hmm. Let me see if D_k can be related to A_{something}?

Alternatively, maybe I can write A_k and D_k as quadratic expressions and see if they can be expressed in terms of linear terms or something else.

Alternatively, maybe I can write A_k and D_k as follows:

Let me note that 4k² + 2k + 1/2 can be written as 4k² + 2k + 1/2. Let me try to complete the square or something.

Alternatively, maybe I can write A_k as (something involving (2k + something)^2). Let me try:

Let me think of 4k² + 2k + 1/2. Let me factor out 4: 4(k² + (1/2)k) + 1/2. Hmm, not sure. Alternatively, maybe think of it as:

Let me think of 4k² + 2k + 1/2. Let me write this as (2k + a)^2 + b. Let me compute:

(2k + a)^2 = 4k² + 4a k + a². So, 4k² + 4a k + a² + b = 4k² + 2k + 1/2. Therefore, equate coefficients:

4a = 2 => a = 1/2. Then, a² + b = 1/2 => (1/2)^2 + b = 1/2 => 1/4 + b = 1/2 => b = 1/4. Therefore, 4k² + 2k + 1/2 = (2k + 1/2)^2 + 1/4.

Similarly, maybe for D_k = 4k² - 6k + 5/2. Let me try the same approach. Let me write it as (2k + c)^2 + d.

(2k + c)^2 = 4k² + 4c k + c². Then, 4k² + 4c k + c² + d = 4k² - 6k + 5/2. Therefore:

4c = -6 => c = -3/2. Then, c² + d = (-3/2)^2 + d = 9/4 + d = 5/2. Therefore, d = 5/2 - 9/4 = 10/4 - 9/4 = 1/4. Therefore, D_k = (2k - 3/2)^2 + 1/4.

So, A_k = (2k + 1/2)^2 + (1/2)^2 and D_k = (2k - 3/2)^2 + (1/2)^2.

Wait, but how does that help? Maybe not directly. Alternatively, perhaps there's a connection between A_k and D_{k - something}? Let me check.

Alternatively, maybe I can write A_k and D_k as terms that can be telescoped when taking the product. Let me think.

Alternatively, maybe I can write A_k and D_k as fractions or something else. Let me compute A_k and D_k:

A_k = 4k² + 2k + 1/2. Let me write this as (8k² + 4k + 1)/2. Similarly, D_k = 4k² - 6k + 5/2 = (8k² - 12k + 5)/2.

So, A_k / D_k = [ (8k² + 4k + 1)/2 ] / [ (8k² - 12k + 5)/2 ] = (8k² + 4k + 1)/(8k² - 12k + 5).

So, the entire product becomes product_{k=1}^n [ (8k² + 4k + 1) / (8k² - 12k + 5) ].

Hmm, maybe I can factor numerator and denominator? Let me try.

First, numerator: 8k² + 4k + 1. Let me check if it factors. The discriminant is 16 - 32 = -16 < 0, so it doesn't factor over real numbers. Similarly for denominator: 8k² - 12k + 5. Discriminant is 144 - 160 = -16 < 0. So they don't factor over real numbers.

Hmm. Maybe there's another approach. Let me think again.

Earlier, I had that each term in the original expression factors into two quadratics, and after cancellation, we have the product of A_k / D_k. Maybe there's a telescoping product here.

Alternatively, maybe I can write A_k and D_k in terms of terms that telescope. Let me think.

Let me note that 8k² + 4k + 1. Let me try to write this as something like (something involving k + 1). Let me check for k and k + 1.

Alternatively, maybe I can relate A_k to D_{k - something}? Let me check.

Let me compute D_{k - 1}:

D_{k - 1} = 8(k - 1)^2 - 12(k - 1) + 5 = 8(k² - 2k + 1) - 12k + 12 + 5 = 8k² - 16k + 8 - 12k + 12 + 5 = 8k² - 28k + 25.

Not sure. Let me check if D_{k - 1} relates to A_k.

Alternatively, maybe I can look for a pattern. Let me compute A_k and D_k for small k and see if there's a telescoping pattern.

Let me take n = 1. Then the original expression is [(2^4 + 1/4)] / [1^4 + 1/4] = (16 + 1/4)/(1 + 1/4) = (65/4)/(5/4) = 65/5 = 13.

Now, according to the product formula, product_{k=1}^1 [A_k / D_k] = A_1 / D_1.

Compute A_1 = 4(1)^2 + 2(1) + 1/2 = 4 + 2 + 0.5 = 6.5 = 13/2.

D_1 = 4(1)^2 - 6(1) + 5/2 = 4 - 6 + 2.5 = 0.5 = 1/2.

Therefore, A_1 / D_1 = (13/2) / (1/2) = 13. Which matches.

So for n=1, the answer is 13.

Let me check n=2. Original expression:

Numerator: (2^4 + 1/4)(4^4 + 1/4) = (16 + 1/4)(256 + 1/4) = (65/4)(1025/4) = (65 * 1025)/16

Denominator: (1^4 + 1/4)(3^4 + 1/4) = (5/4)(81 + 1/4) = (5/4)(325/4) = (5 * 325)/16

Therefore, the ratio is (65 * 1025) / (5 * 325) = (13 * 5 * 1025) / (5 * 65) ) = (13 * 1025)/65. Wait, 65 is 5*13. So 13 * 1025 / (5*13) ) = 1025 / 5 = 205.

Alternatively, compute via the product formula: product_{k=1}^2 [A_k / D_k] = (A_1 / D_1) * (A_2 / D_2).

A_1 / D_1 = 13 as before.

Compute A_2 = 4*(2)^2 + 2*(2) + 1/2 = 16 + 4 + 0.5 = 20.5 = 41/2.

D_2 = 4*(2)^2 - 6*(2) + 5/2 = 16 - 12 + 2.5 = 6.5 = 13/2.

Therefore, A_2 / D_2 = (41/2) / (13/2) = 41/13.

Therefore, total product is 13 * (41/13) = 41. Which is different from 205? Wait, no, wait. Wait, earlier I thought the ratio was 205, but according to this, it's 41? Wait, there must be a mistake here.

Wait, let me recalculate the original expression for n=2.

Original expression:

Numerator: (2^4 + 1/4)(4^4 + 1/4) = (16 + 0.25)(256 + 0.25) = 16.25 * 256.25.

Denominator: (1^4 + 1/4)(3^4 + 1/4) = (1 + 0.25)(81 + 0.25) = 1.25 * 81.25.

Compute numerator: 16.25 * 256.25. Let me compute 16 * 256.25 = 4100, and 0.25 * 256.25 = 64.0625, so total is 4100 + 64.0625 = 4164.0625.

Denominator: 1.25 * 81.25. 1 * 81.25 = 81.25, 0.25 * 81.25 = 20.3125, total is 101.5625.

Therefore, ratio is 4164.0625 / 101.5625. Let me compute that.

Divide numerator and denominator by 1.25: numerator becomes 4164.0625 / 1.25 = 3331.25, denominator becomes 101.5625 / 1.25 = 81.25.

Then 3331.25 / 81.25. Let me divide numerator and denominator by 1.25 again: numerator becomes 3331.25 / 1.25 = 2665, denominator becomes 81.25 / 1.25 = 65.

So 2665 / 65. Let me divide 2665 by 65. 65*40 = 2600, so 2665 - 2600 = 65, so total is 40 + 1 = 41.

Ah, so the ratio is 41. Which matches the product formula. So earlier when I thought I had 205, I must have miscalculated. So the product formula gives 41, which is correct.

So for n=1, answer is 13; for n=2, answer is 41. Let me check n=3. Maybe I can see a pattern.

But before that, let me think again. So the product is product_{k=1}^n [A_k / D_k] = product_{k=1}^n [ (8k² + 4k + 1) / (8k² - 12k + 5) ].

Let me try to write numerator and denominator in terms of something. Let me note that 8k² + 4k + 1 can be written as 8k² + 4k + 1. Let me check if this is similar to 8(k + something)^2 + something.

Alternatively, maybe I can write numerator as 8k² + 4k + 1 = 8k² + 4k + 1. Let me think of this as 8k² + 4k + 1 = 8k² + 4k + 1.

Alternatively, maybe I can relate numerator of term k to denominator of term k + something. Let me check for k and k + 1.

Let me compute denominator of term k: 8k² - 12k + 5. Let me compute numerator of term k + 1: 8(k + 1)^2 + 4(k + 1) + 1 = 8(k² + 2k + 1) + 4k + 4 + 1 = 8k² + 16k + 8 + 4k + 5 = 8k² + 20k + 13.

Not sure if that relates to denominator of term k.

Alternatively, let me check if denominator of term k is equal to numerator of term k - something. Let me check denominator of term k: 8k² - 12k + 5. Let me check numerator of term k - 1: 8(k - 1)^2 + 4(k - 1) + 1 = 8(k² - 2k + 1) + 4k - 4 + 1 = 8k² - 16k + 8 + 4k - 3 = 8k² - 12k + 5.

Oh! Wait! That's exactly the denominator of term k.

So, denominator of term k is equal to numerator of term k - 1.

Therefore, D_k = numerator of term (k - 1).

Therefore, the product becomes:

product_{k=1}^n [A_k / D_k] = product_{k=1}^n [A_k / A_{k - 1} ] where A_{k - 1} is the numerator of term k - 1.

But note that when k=1, D_1 = numerator of term 0. Wait, but term 0? Let me check.

Wait, for k=1, D_1 = 8(1)^2 - 12(1) + 5 = 8 - 12 + 5 = 1. Which is equal to numerator of term k - 1 = term 0. But term 0 would be A_0 = 8(0)^2 + 4(0) + 1 = 1. So yes, D_1 = A_0.

Therefore, the product becomes:

product_{k=1}^n [A_k / A_{k - 1} ] = A_n / A_0.

Because it's telescoping: A_1 / A_0 * A_2 / A_1 * A_3 / A_2 * ... * A_n / A_{n - 1} } = A_n / A_0.

Therefore, the entire expression simplifies to A_n / A_0.

Since A_0 = 8(0)^2 + 4(0) + 1 = 1. Therefore, the product is A_n / 1 = A_n.

Therefore, the original expression simplifies to A_n, which is 8n² + 4n + 1.

Wait, let me check with n=1: A_1 = 8(1)^2 + 4(1) + 1 = 8 + 4 + 1 = 13. Which matches. For n=2, A_2 = 8(4) + 8 + 1 = 32 + 8 + 1 = 41. Which matches. For n=3, A_3 = 8(9) + 12 + 1 = 72 + 12 + 1 = 85. Let me check with original expression for n=3.

Original expression for n=3:

Numerator: (2^4 + 1/4)(4^4 + 1/4)(6^4 + 1/4) = (16 + 0.25)(256 + 0.25)(1296 + 0.25) = 16.25 * 256.25 * 1296.25

Denominator: (1^4 + 1/4)(3^4 + 1/4)(5^4 + 1/4) = (1.25)(81.25)(625.25)

But instead of computing this, since we have the formula A_n = 8n² + 4n + 1, for n=3, it's 8*9 + 12 + 1 = 72 + 13 = 85.

Alternatively, using the telescoping product, since the product is A_n / A_0 = A_n / 1 = A_n. Therefore, the answer is 8n² + 4n + 1.

But let me check with n=3. Let me compute the product formula:

product_{k=1}^3 [A_k / D_k] = A_3 / A_0 = 85 / 1 = 85.

Alternatively, compute the original expression:

Numerator: (2^4 + 1/4)(4^4 + 1/4)(6^4 + 1/4) = (65/4)(1025/4)(1296 + 1/4) = (65/4)(1025/4)(5185/4).

Denominator: (1^4 + 1/4)(3^4 + 1/4)(5^4 + 1/4) = (5/4)(325/4)(625 + 1/4) = (5/4)(325/4)(2501/4).

So the ratio is [65 * 1025 * 5185] / [5 * 325 * 2501] * (4^3 / 4^3) = [65 * 1025 * 5185] / [5 * 325 * 2501].

Simplify step by step:

First, 65 / 5 = 13.

Then, 1025 / 325 = 1025 / 325 = divide numerator and denominator by 25: 41 / 13.

Then, 5185 / 2501. Let me check if 2501 divides into 5185. Let me compute 2501 * 2 = 5002. 5185 - 5002 = 183. So 5185 = 2501 * 2 + 183. Not sure. Wait, but according to our previous conclusion, the ratio should be 85. Let me check:

If the ratio is 13 * (41/13) * (5185 / 2501) = 41 * (5185 / 2501).

Now, compute 5185 / 2501. Let me check if 2501 * 2 = 5002, 5185 - 5002 = 183. Then 2501 * 2 + 183 = 5185. Not sure. Wait, but according to our telescoping product, it should be 85. Let me check 85 * 2501 = ?

Well, 85 * 2501 = 85 * 2500 + 85 * 1 = 212500 + 85 = 212585.

Now, 41 * 5185 = 41 * 5000 + 41 * 185 = 205000 + 7585 = 212585.

Yes! Therefore, 41 * 5185 = 212585 = 85 * 2501. Therefore, the ratio is 85. Which matches.

Therefore, the formula seems to hold. Therefore, the original expression simplifies to A_n = 8n² + 4n + 1.

But let me check if this can be simplified further or written in another form.

Alternatively, maybe factor 8n² + 4n + 1. Let me check discriminant: 16 - 32 = -16. So it doesn't factor over real numbers. So maybe that's the simplest form.

But let me check if there's another way to write it. For example, 8n² + 4n + 1. Let me think.

Alternatively, maybe write it as (2n + 1)^2 + (2n)^2? Let me check:

(2n + 1)^2 = 4n² + 4n + 1. Then adding (2n)^2 = 4n². Total is 8n² + 4n + 1. Yes! So 8n² + 4n + 1 = (2n + 1)^2 + (2n)^2.

But not sure if that helps. Alternatively, maybe write it as something else.

Alternatively, maybe write it as 4(2n² + n) + 1. But again, not sure.

Alternatively, maybe check if there's a connection with the original problem.

But since the problem says "simplify the expression", and 8n² + 4n + 1 is a simplified form, unless there's a more elegant expression.

Alternatively, maybe write it as (something). Let me think.

Alternatively, note that 8n² + 4n + 1 = 4(2n² + n) + 1. But again, not helpful.

Alternatively, maybe write it as (something involving (2n + 1)^2).

But since the problem says "simplify", and given that the answer is a quadratic in n, I think 8n² + 4n + 1 is the simplified form.

But let me check for n=1, 8(1)^2 + 4(1) + 1 = 13, which is correct. For n=2, 8(4) + 8 + 1 = 32 + 8 + 1 = 41, which is correct. For n=3, 8(9) + 12 + 1 = 72 + 13 = 85, which is correct.

Therefore, the simplified expression is 8n² + 4n + 1.

But let me check if there's another way to write this. For example, maybe factor out something?

Alternatively, note that 8n² + 4n + 1 = 4(2n² + n) + 1. But not helpful.

Alternatively, maybe write it as (2n + 1)^2 + (2n)^2. Wait, as I did before.

But maybe the problem expects a different form? Let me think again.

Wait, going back to the original problem, the expression was:

[(2^4 + 1/4)(4^4 + 1/4)...(2n)^4 + 1/4] / [(1^4 + 1/4)(3^4 + 1/4)...(2n - 1)^4 + 1/4]

And we found that this simplifies to 8n² + 4n + 1.

Alternatively, maybe there's a way to write this as (something with (2n + 1)^2 or similar).

Alternatively, maybe check if 8n² + 4n + 1 can be written as (something).

Alternatively, maybe think of the original telescoping product.

But since we have already confirmed with multiple examples and the telescoping logic, I think 8n² + 4n + 1 is the correct simplified form.

Therefore, the answer is 8n² + 4n + 1.

But let me check once more for n=1, 2, 3.

n=1: 8(1) + 4(1) + 1 = 13. Correct.

n=2: 8(4) + 4(2) + 1 = 32 + 8 + 1 = 41. Correct.

n=3: 8(9) + 4(3) + 1 = 72 + 12 + 1 = 85. Correct.

Therefore, I think this is the correct answer.

But let me check if there's an alternative expression. For example, maybe (2n + 1)^2 + (2n)^2?

(2n + 1)^2 + (2n)^2 = 4n² + 4n + 1 + 4n² = 8n² + 4n + 1. Yes! So 8n² + 4n + 1 = (2n)^2 + (2n + 1)^2.

But maybe the problem expects this form? Or maybe another form?

Alternatively, maybe write it as (something else).

Alternatively, maybe think of the original expression.

But since the problem says "simplify", and 8n² + 4n + 1 is a simplified polynomial, I think that's acceptable.

Alternatively, maybe factor it as something? But since it doesn't factor over integers, I think that's the simplest.

Therefore, the answer is 8n² + 4n + 1.

But let me check once again the telescoping logic.

We had that D_k = A_{k - 1}, so the product becomes A_n / A_0. Since A_0 = 1, then the product is A_n.

Yes, and A_n = 8n² + 4n + 1.

Therefore, the final answer is 8n² + 4n + 1.

But let me check if there's a different approach.

Alternatively, going back to the original factorization.

We had that each term in the numerator and denominator factors into two quadratics, and after cancellation, we had the product of A_k / D_k, which telescopes to A_n.

Alternatively, maybe there's another way to see this.

But given that multiple examples check out and the telescoping logic is sound, I think this is correct.

Therefore, the simplified expression is 8n² + 4n + 1.

**Final Answer**
\boxed{8n^2 + 4n + 1}

To simplify the expression

$$
\frac{(2^4 + \frac{1}{4})(4^4 + \frac{1}{4}) \dotsm [(2n)^4 + \frac{1}{4}]}{(1^4 + \frac{1}{4})(3^4 + \frac{1}{4}) \dotsm [(2n - 1)^4 + \frac{1}{4}]}
$$

we begin by recognizing a pattern in the terms. Each term in the numerator and denominator is of the form $ x^4 + \frac{1}{4} $, which can be factored using the **Sophie Germain identity**:

$$
x^4 + 4y^4 = (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2)
$$

Setting $ y = \frac{1}{2} $, we get:

$$
x^4 + \frac{1}{4} = (x^2 + x + \frac{1}{2})(x^2 - x + \frac{1}{2})
$$

---

### Step 1: Factor Each Term

Let’s apply this identity to both the numerator and denominator:

- **Numerator terms** (even $ x = 2k $):

$$
(2k)^4 + \frac{1}{4} = (4k^2 + 2k + \frac{1}{2})(4k^2 - 2k + \frac{1}{2})
$$

- **Denominator terms** (odd $ x = 2k - 1 $):

$$
(2k - 1)^4 + \frac{1}{4} = (4k^2 - 2k + \frac{1}{2})(4k^2 - 6k + \frac{5}{2})
$$

---

### Step 2: Write the Full Expression

Now, the full expression becomes:

$$
\frac{\prod_{k=1}^n (4k^2 + 2k + \frac{1}{2})(4k^2 - 2k + \frac{1}{2})}{\prod_{k=1}^n (4k^2 - 2k + \frac{1}{2})(4k^2 - 6k + \frac{5}{2})}
$$

Notice that the second factor in the numerator, $ 4k^2 - 2k + \frac{1}{2} $, is the same as the first factor in the denominator, $ 4k^2 - 2k + \frac{1}{2} $. So, these terms cancel out across the product.

This leaves us with:

$$
\frac{\prod_{k=1}^n (4k^2 + 2k + \frac{1}{2})}{\prod_{k=1}^n (4k^2 - 6k + \frac{5}{2})}
$$

---

### Step 3: Simplify the Remaining Product

Let’s define:

- $ A_k = 4k^2 + 2k + \frac{1}{2} $
- $ D_k = 4k^2 - 6k + \frac{5}{2} $

Then the expression becomes:

$$
\prod_{k=1}^n \frac{A_k}{D_k}
$$

Now, observe that:

$$
D_k = A_{k-1}
$$

This is because:

$$
A_{k-1} = 4(k-1)^2 + 2(k-1) + \frac{1}{2} = 4k^2 - 8k + 4 + 2k - 2 + \frac{1}{2} = 4k^2 - 6k + \frac{5}{2}
$$

So, the product becomes:

$$
\prod_{k=1}^n \frac{A_k}{A_{k-1}} = \frac{A_n}{A_0}
$$

Since $ A_0 = 4(0)^2 + 2(0) + \frac{1}{2} = \frac{1}{2} $, we have:

$$
\frac{A_n}{A_0} = \frac{A_n}{\frac{1}{2}} = 2A_n
$$

But wait! Actually, we had:

$$
\prod_{k=1}^n \frac{A_k}{D_k} = \prod_{k=1}^n \frac{A_k}{A_{k-1}} = \frac{A_n}{A_0}
$$

So the final expression is:

$$
\frac{A_n}{A_0} = \frac{8n^2 + 4n + 1}{1} = 8n^2 + 4n + 1
$$

---

### Final Answer

$$
\boxed{8n^2 + 4n + 1}
$$

[Assistant]
Explanation:
Okay, so I need to simplify this big fraction where the numerator is the product of terms like (2^4 + 1/4), (4^4 + 1/4), all the way up to (2n)^4 + 1/4, and the denominator is the product of similar terms but with odd numbers: (1^4 + 1/4), (3^4 + 1/4), ..., up to (2n - 1)^4 + 1/4.

First, let me write down the expression more clearly:

$$
\frac{(2^4 + \frac{1}{4})(4^4 + \frac{1}{4}) \dotsm [(2n)^4 + \frac{1}{4}]}{(1^4 + \frac{1}{4})(3^4 + \frac{1}{4}) \dotsm [(2n - 1)^4 + \frac{1}{4}]}
$$

So, the numerator is the product of (2k)^4 + 1/4 for k from 1 to n, and the denominator is the product of (2k - 1)^4 + 1/4 for k from 1 to n.

Hmm, maybe I can first try to simplify each term in the numerator and denominator. Let me take a general term, say, for a number x, x^4 + 1/4. Maybe there's a way to factor this expression?

I remember that x^4 + 4y^4 can be factored using the Sophie Germain identity: x^4 + 4y^4 = (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2). Wait, let me check that.

Yes, the Sophie Germain identity is x^4 + 4y^4 = (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2). So if I can write x^4 + 1/4 in terms of this identity, maybe?

Let me see. Let me set 4y^4 = 1/4. Then y^4 = 1/16, so y = 1/2. Therefore, x^4 + 1/4 = x^4 + 4*(1/2)^4. So that would be equal to (x^2 + 2x*(1/2) + 2*(1/2)^2)(x^2 - 2x*(1/2) + 2*(1/2)^2). Let me compute that.

First term: x^2 + 2x*(1/2) + 2*(1/2)^2 = x^2 + x + 2*(1/4) = x^2 + x + 1/2.

Second term: x^2 - 2x*(1/2) + 2*(1/2)^2 = x^2 - x + 1/2.

Therefore, x^4 + 1/4 factors into (x^2 + x + 1/2)(x^2 - x + 1/2).

So, for each term in the numerator and denominator, we can factor them like that. Let me check with a specific example. Let me take x = 2. Then 2^4 + 1/4 = 16 + 1/4 = 65/4. On the other hand, (2^2 + 2 + 1/2)(2^2 - 2 + 1/2) = (4 + 2 + 0.5)(4 - 2 + 0.5) = (6.5)(2.5) = 16.25, which is 65/4. Correct. So that works.

Similarly, for x = 1: 1^4 + 1/4 = 1 + 1/4 = 5/4. Factoring: (1 + 1 + 1/2)(1 - 1 + 1/2) = (2.5)(0.5) = 1.25 = 5/4. Correct.

So, this factorization works. Therefore, each term in the numerator and denominator can be written as a product of two quadratics.

Therefore, the entire expression can be written as:

Numerator: product_{k=1}^n [( (2k)^2 + 2k + 1/2 ) * ( (2k)^2 - 2k + 1/2 ) ]

Denominator: product_{k=1}^n [ ( (2k - 1)^2 + (2k - 1) + 1/2 ) * ( (2k - 1)^2 - (2k - 1) + 1/2 ) ]

So, if I can write each term in numerator and denominator as such, maybe there is some telescoping or cancellation when taking the ratio.

Let me compute the terms in the numerator and denominator separately. Let me first compute the factors for the numerator terms. Let me denote for each term in the numerator, which is (2k)^4 + 1/4, which factors into:

[( (2k)^2 + 2*(2k) + 1/2 ) * ( (2k)^2 - 2*(2k) + 1/2 ) ] ?

Wait, no. Wait, earlier we had x^4 + 1/4 = (x^2 + x + 1/2)(x^2 - x + 1/2). So for x = 2k, it would be:

( (2k)^2 + (2k) + 1/2 ) * ( (2k)^2 - (2k) + 1/2 )

Wait, but earlier when I did x^4 + 1/4, the factors were (x^2 + x + 1/2)(x^2 - x + 1/2). So yes, that's correct. So for x = 2k, the factors are:

( (2k)^2 + 2k + 1/2 ) and ( (2k)^2 - 2k + 1/2 )

Similarly, for the denominator terms, which are (2k - 1)^4 + 1/4, so x = 2k - 1, so factors are:

( (2k - 1)^2 + (2k - 1) + 1/2 ) * ( (2k - 1)^2 - (2k - 1) + 1/2 )

So, perhaps if I can write these factors in terms of something that telescopes when taking the product over k.

Alternatively, maybe I can write each quadratic term as something else. Let me compute them:

First, let me compute the first factor in the numerator term:

( (2k)^2 + 2k + 1/2 ) = 4k^2 + 2k + 1/2. Similarly, the second factor is 4k^2 - 2k + 1/2.

Similarly, for the denominator terms:

First factor: ( (2k - 1)^2 + (2k - 1) + 1/2 )

Let me compute that:

(2k - 1)^2 = 4k^2 - 4k + 1. Then adding (2k - 1) gives 4k^2 - 4k + 1 + 2k - 1 = 4k^2 - 2k. Then adding 1/2 gives 4k^2 - 2k + 1/2.

Similarly, the second factor for the denominator term is:

( (2k - 1)^2 - (2k - 1) + 1/2 )

Compute that:

(2k - 1)^2 - (2k - 1) + 1/2 = 4k^2 - 4k + 1 - 2k + 1 + 1/2 = Wait, let me do it step by step.

First, (2k - 1)^2 = 4k^2 - 4k + 1. Then subtract (2k - 1): 4k^2 - 4k + 1 - 2k + 1 = 4k^2 - 6k + 2. Then add 1/2: 4k^2 - 6k + 2 + 1/2 = 4k^2 - 6k + 5/2. Wait, that seems complicated. Wait, maybe I made a mistake here.

Wait, let me redo that. The second factor for the denominator term is:

( (2k - 1)^2 - (2k - 1) + 1/2 )

So:

First term: (2k - 1)^2 = 4k² - 4k + 1

Second term: - (2k - 1) = -2k + 1

Third term: + 1/2

So adding them all together:

4k² - 4k + 1 - 2k + 1 + 1/2 = 4k² - 6k + 2 + 1/2 = 4k² - 6k + 5/2.

Hmm, okay. So that's the second factor for the denominator term.

But maybe there's a better way to write these expressions. Let me check if these quadratic terms can be related to something else.

Alternatively, maybe I can write the numerator and denominator terms as products of terms that telescope when taking the ratio. Let me think.

Let me first look at the numerator factors. For each term in the numerator, we have two factors:

A_k = 4k² + 2k + 1/2 and B_k = 4k² - 2k + 1/2.

Similarly, for the denominator terms, each term is:

C_k = 4k² - 2k + 1/2 and D_k = 4k² - 6k + 5/2? Wait, no. Wait, earlier for the denominator terms, the first factor was 4k² - 2k + 1/2, which is same as B_k? Wait, no. Wait, for the denominator term, the first factor was:

( (2k - 1)^2 + (2k - 1) + 1/2 ) = 4k² - 2k + 1/2. Wait, earlier when I calculated that, I had:

(2k - 1)^2 + (2k - 1) + 1/2 = 4k² - 4k + 1 + 2k - 1 + 1/2 = 4k² - 2k + 1/2. Yes, so that is equal to 4k² - 2k + 1/2. Which is same as B_k? Wait, B_k was 4k² - 2k + 1/2. Yes! So the first factor in the denominator term is equal to B_k.

Wait, but the denominator term is (2k - 1)^4 + 1/4, which factors into ( (2k - 1)^2 + (2k - 1) + 1/2 ) * ( (2k - 1)^2 - (2k - 1) + 1/2 ) = C_k * D_k, where C_k = 4k² - 2k + 1/2 and D_k = 4k² - 6k + 5/2.

But the numerator term is (2k)^4 + 1/4 = A_k * B_k, where A_k = 4k² + 2k + 1/2 and B_k = 4k² - 2k + 1/2.

So, notice that the B_k in the numerator is same as C_k in the denominator. Therefore, when we take the ratio of numerator over denominator, each term in the numerator has a B_k, and each term in the denominator has a C_k. But since C_k is same as B_k, perhaps there is some cancellation? Wait, but the denominator has terms for k from 1 to n, and the numerator also has terms for k from 1 to n.

Wait, but let me think. Let me write the entire product:

Numerator: product_{k=1}^n [A_k * B_k] = product_{k=1}^n A_k * product_{k=1}^n B_k

Denominator: product_{k=1}^n [C_k * D_k] = product_{k=1}^n C_k * product_{k=1}^n D_k

But since C_k = B_k, then product_{k=1}^n C_k = product_{k=1}^n B_k. Therefore, the ratio becomes:

[ product_{k=1}^n A_k * product_{k=1}^n B_k ] / [ product_{k=1}^n B_k * product_{k=1}^n D_k ] = [ product_{k=1}^n A_k ] / [ product_{k=1}^n D_k ]

Therefore, the ratio simplifies to the product of A_k divided by the product of D_k.

So now, the problem reduces to computing:

Product_{k=1}^n [A_k / D_k] where A_k = 4k² + 2k + 1/2 and D_k = 4k² - 6k + 5/2.

Wait, but maybe I can write A_k and D_k in terms of something else. Let me check if there's a relation between A_k and D_{k} or D_{k} and A_{k+1} or something like that.

Let me compute A_k and D_k:

A_k = 4k² + 2k + 1/2

D_k = 4k² - 6k + 5/2

Hmm. Let me see if D_k can be related to A_{something}?

Alternatively, maybe I can write A_k and D_k as quadratic expressions and see if they can be expressed in terms of linear terms or something else.

Alternatively, maybe I can write A_k and D_k as follows:

Let me note that 4k² + 2k + 1/2 can be written as 4k² + 2k + 1/2. Let me try to complete the square or something.

Alternatively, maybe I can write A_k as (something involving (2k + something)^2). Let me try:

Let me think of 4k² + 2k + 1/2. Let me factor out 4: 4(k² + (1/2)k) + 1/2. Hmm, not sure. Alternatively, maybe think of it as:

Let me think of 4k² + 2k + 1/2. Let me write this as (2k + a)^2 + b. Let me compute:

(2k + a)^2 = 4k² + 4a k + a². So, 4k² + 4a k + a² + b = 4k² + 2k + 1/2. Therefore, equate coefficients:

4a = 2 => a = 1/2. Then, a² + b = 1/2 => (1/2)^2 + b = 1/2 => 1/4 + b = 1/2 => b = 1/4. Therefore, 4k² + 2k + 1/2 = (2k + 1/2)^2 + 1/4.

Similarly, maybe for D_k = 4k² - 6k + 5/2. Let me try the same approach. Let me write it as (2k + c)^2 + d.

(2k + c)^2 = 4k² + 4c k + c². Then, 4k² + 4c k + c² + d = 4k² - 6k + 5/2. Therefore:

4c = -6 => c = -3/2. Then, c² + d = (-3/2)^2 + d = 9/4 + d = 5/2. Therefore, d = 5/2 - 9/4 = 10/4 - 9/4 = 1/4. Therefore, D_k = (2k - 3/2)^2 + 1/4.

So, A_k = (2k + 1/2)^2 + (1/2)^2 and D_k = (2k - 3/2)^2 + (1/2)^2.

Wait, but how does that help? Maybe not directly. Alternatively, perhaps there's a connection between A_k and D_{k - something}? Let me check.

Alternatively, maybe I can write A_k and D_k as terms that can be telescoped when taking the product. Let me think.

Alternatively, maybe I can write A_k and D_k as fractions or something else. Let me compute A_k and D_k:

A_k = 4k² + 2k + 1/2. Let me write this as (8k² + 4k + 1)/2. Similarly, D_k = 4k² - 6k + 5/2 = (8k² - 12k + 5)/2.

So, A_k / D_k = [ (8k² + 4k + 1)/2 ] / [ (8k² - 12k + 5)/2 ] = (8k² + 4k + 1)/(8k² - 12k + 5).

So, the entire product becomes product_{k=1}^n [ (8k² + 4k + 1) / (8k² - 12k + 5) ].

Hmm, maybe I can factor numerator and denominator? Let me try.

First, numerator: 8k² + 4k + 1. Let me check if it factors. The discriminant is 16 - 32 = -16 < 0, so it doesn't factor over real numbers. Similarly for denominator: 8k² - 12k + 5. Discriminant is 144 - 160 = -16 < 0. So they don't factor over real numbers.

Hmm. Maybe there's another approach. Let me think again.

Earlier, I had that each term in the original expression factors into two quadratics, and after cancellation, we have the product of A_k / D_k. Maybe there's a telescoping product here.

Alternatively, maybe I can write A_k and D_k in terms of terms that telescope. Let me think.

Let me note that 8k² + 4k + 1. Let me try to write this as something like (something involving k + 1). Let me check for k and k + 1.

Alternatively, maybe I can relate A_k to D_{k - something}? Let me check.

Let me compute D_{k - 1}:

D_{k - 1} = 8(k - 1)^2 - 12(k - 1) + 5 = 8(k² - 2k + 1) - 12k + 12 + 5 = 8k² - 16k + 8 - 12k + 12 + 5 = 8k² - 28k + 25.

Not sure. Let me check if D_{k - 1} relates to A_k.

Alternatively, maybe I can look for a pattern. Let me compute A_k and D_k for small k and see if there's a telescoping pattern.

Let me take n = 1. Then the original expression is [(2^4 + 1/4)] / [1^4 + 1/4] = (16 + 1/4)/(1 + 1/4) = (65/4)/(5/4) = 65/5 = 13.

Now, according to the product formula, product_{k=1}^1 [A_k / D_k] = A_1 / D_1.

Compute A_1 = 4(1)^2 + 2(1) + 1/2 = 4 + 2 + 0.5 = 6.5 = 13/2.

D_1 = 4(1)^2 - 6(1) + 5/2 = 4 - 6 + 2.5 = 0.5 = 1/2.

Therefore, A_1 / D_1 = (13/2) / (1/2) = 13. Which matches.

So for n=1, the answer is 13.

Let me check n=2. Original expression:

Numerator: (2^4 + 1/4)(4^4 + 1/4) = (16 + 1/4)(256 + 1/4) = (65/4)(1025/4) = (65 * 1025)/16

Denominator: (1^4 + 1/4)(3^4 + 1/4) = (5/4)(81 + 1/4) = (5/4)(325/4) = (5 * 325)/16

Therefore, the ratio is (65 * 1025) / (5 * 325) = (13 * 5 * 1025) / (5 * 65) ) = (13 * 1025)/65. Wait, 65 is 5*13. So 13 * 1025 / (5*13) ) = 1025 / 5 = 205.

Alternatively, compute via the product formula: product_{k=1}^2 [A_k / D_k] = (A_1 / D_1) * (A_2 / D_2).

A_1 / D_1 = 13 as before.

Compute A_2 = 4*(2)^2 + 2*(2) + 1/2 = 16 + 4 + 0.5 = 20.5 = 41/2.

D_2 = 4*(2)^2 - 6*(2) + 5/2 = 16 - 12 + 2.5 = 6.5 = 13/2.

Therefore, A_2 / D_2 = (41/2) / (13/2) = 41/13.

Therefore, total product is 13 * (41/13) = 41. Which is different from 205? Wait, no, wait. Wait, earlier I thought the ratio was 205, but according to this, it's 41? Wait, there must be a mistake here.

Wait, let me recalculate the original expression for n=2.

Original expression:

Numerator: (2^4 + 1/4)(4^4 + 1/4) = (16 + 0.25)(256 + 0.25) = 16.25 * 256.25.

Denominator: (1^4 + 1/4)(3^4 + 1/4) = (1 + 0.25)(81 + 0.25) = 1.25 * 81.25.

Compute numerator: 16.25 * 256.25. Let me compute 16 * 256.25 = 4100, and 0.25 * 256.25 = 64.0625, so total is 4100 + 64.0625 = 4164.0625.

Denominator: 1.25 * 81.25. 1 * 81.25 = 81.25, 0.25 * 81.25 = 20.3125, total is 101.5625.

Therefore, ratio is 4164.0625 / 101.5625. Let me compute that.

Divide numerator and denominator by 1.25: numerator becomes 4164.0625 / 1.25 = 3331.25, denominator becomes 101.5625 / 1.25 = 81.25.

Then 3331.25 / 81.25. Let me divide numerator and denominator by 1.25 again: numerator becomes 3331.25 / 1.25 = 2665, denominator becomes 81.25 / 1.25 = 65.

So 2665 / 65. Let me divide 2665 by 65. 65*40 = 2600, so 2665 - 2600 = 65, so total is 40 + 1 = 41.

Ah, so the ratio is 41. Which matches the product formula. So earlier when I thought I had 205, I must have miscalculated. So the product formula gives 41, which is correct.

So for n=1, answer is 13; for n=2, answer is 41. Let me check n=3. Maybe I can see a pattern.

But before that, let me think again. So the product is product_{k=1}^n [A_k / D_k] = product_{k=1}^n [ (8k² + 4k + 1) / (8k² - 12k + 5) ].

Let me try to write numerator and denominator in terms of something. Let me note that 8k² + 4k + 1 can be written as 8k² + 4k + 1. Let me check if this is similar to 8(k + something)^2 + something.

Alternatively, maybe I can write numerator as 8k² + 4k + 1 = 8k² + 4k + 1. Let me think of this as 8k² + 4k + 1 = 8k² + 4k + 1.

Alternatively, maybe I can relate numerator of term k to denominator of term k + something. Let me check for k and k + 1.

Let me compute denominator of term k: 8k² - 12k + 5. Let me compute numerator of term k + 1: 8(k + 1)^2 + 4(k + 1) + 1 = 8(k² + 2k + 1) + 4k + 4 + 1 = 8k² + 16k + 8 + 4k + 5 = 8k² + 20k + 13.

Not sure if that relates to denominator of term k.

Alternatively, let me check if denominator of term k is equal to numerator of term k - something. Let me check denominator of term k: 8k² - 12k + 5. Let me check numerator of term k - 1: 8(k - 1)^2 + 4(k - 1) + 1 = 8(k² - 2k + 1) + 4k - 4 + 1 = 8k² - 16k + 8 + 4k - 3 = 8k² - 12k + 5.

Oh! Wait! That's exactly the denominator of term k.

So, denominator of term k is equal to numerator of term k - 1.

Therefore, D_k = numerator of term (k - 1).

Therefore, the product becomes:

product_{k=1}^n [A_k / D_k] = product_{k=1}^n [A_k / A_{k - 1} ] where A_{k - 1} is the numerator of term k - 1.

But note that when k=1, D_1 = numerator of term 0. Wait, but term 0? Let me check.

Wait, for k=1, D_1 = 8(1)^2 - 12(1) + 5 = 8 - 12 + 5 = 1. Which is equal to numerator of term k - 1 = term 0. But term 0 would be A_0 = 8(0)^2 + 4(0) + 1 = 1. So yes, D_1 = A_0.

Therefore, the product becomes:

product_{k=1}^n [A_k / A_{k - 1} ] = A_n / A_0.

Because it's telescoping: A_1 / A_0 * A_2 / A_1 * A_3 / A_2 * ... * A_n / A_{n - 1} } = A_n / A_0.

Therefore, the entire expression simplifies to A_n / A_0.

Since A_0 = 8(0)^2 + 4(0) + 1 = 1. Therefore, the product is A_n / 1 = A_n.

Therefore, the original expression simplifies to A_n, which is 8n² + 4n + 1.

Wait, let me check with n=1: A_1 = 8(1)^2 + 4(1) + 1 = 8 + 4 + 1 = 13. Which matches. For n=2, A_2 = 8(4) + 8 + 1 = 32 + 8 + 1 = 41. Which matches. For n=3, A_3 = 8(9) + 12 + 1 = 72 + 12 + 1 = 85. Let me check with original expression for n=3.

Original expression for n=3:

Numerator: (2^4 + 1/4)(4^4 + 1/4)(6^4 + 1/4) = (16 + 0.25)(256 + 0.25)(1296 + 0.25) = 16.25 * 256.25 * 1296.25

Denominator: (1^4 + 1/4)(3^4 + 1/4)(5^4 + 1/4) = (1.25)(81.25)(625.25)

But instead of computing this, since we have the formula A_n = 8n² + 4n + 1, for n=3, it's 8*9 + 12 + 1 = 72 + 13 = 85.

Alternatively, using the telescoping product, since the product is A_n / A_0 = A_n / 1 = A_n. Therefore, the answer is 8n² + 4n + 1.

But let me check with n=3. Let me compute the product formula:

product_{k=1}^3 [A_k / D_k] = A_3 / A_0 = 85 / 1 = 85.

Alternatively, compute the original expression:

Numerator: (2^4 + 1/4)(4^4 + 1/4)(6^4 + 1/4) = (65/4)(1025/4)(1296 + 1/4) = (65/4)(1025/4)(5185/4).

Denominator: (1^4 + 1/4)(3^4 + 1/4)(5^4 + 1/4) = (5/4)(325/4)(625 + 1/4) = (5/4)(325/4)(2501/4).

So the ratio is [65 * 1025 * 5185] / [5 * 325 * 2501] * (4^3 / 4^3) = [65 * 1025 * 5185] / [5 * 325 * 2501].

Simplify step by step:

First, 65 / 5 = 13.

Then, 1025 / 325 = 1025 / 325 = divide numerator and denominator by 25: 41 / 13.

Then, 5185 / 2501. Let me check if 2501 divides into 5185. Let me compute 2501 * 2 = 5002. 5185 - 5002 = 183. So 5185 = 2501 * 2 + 183. Not sure. Wait, but according to our previous conclusion, the ratio should be 85. Let me check:

If the ratio is 13 * (41/13) * (5185 / 2501) = 41 * (5185 / 2501).

Now, compute 5185 / 2501. Let me check if 2501 * 2 = 5002, 5185 - 5002 = 183. Then 2501 * 2 + 183 = 5185. Not sure. Wait, but according to our telescoping product, it should be 85. Let me check 85 * 2501 = ?

Well, 85 * 2501 = 85 * 2500 + 85 * 1 = 212500 + 85 = 212585.

Now, 41 * 5185 = 41 * 5000 + 41 * 185 = 205000 + 7585 = 212585.

Yes! Therefore, 41 * 5185 = 212585 = 85 * 2501. Therefore, the ratio is 85. Which matches.

Therefore, the formula seems to hold. Therefore, the original expression simplifies to A_n = 8n² + 4n + 1.

But let me check if this can be simplified further or written in another form.

Alternatively, maybe factor 8n² + 4n + 1. Let me check discriminant: 16 - 32 = -16. So it doesn't factor over real numbers. So maybe that's the simplest form.

But let me check if there's another way to write it. For example, 8n² + 4n + 1. Let me think.

Alternatively, maybe write it as (2n + 1)^2 + (2n)^2? Let me check:

(2n + 1)^2 = 4n² + 4n + 1. Then adding (2n)^2 = 4n². Total is 8n² + 4n + 1. Yes! So 8n² + 4n + 1 = (2n + 1)^2 + (2n)^2.

But not sure if that helps. Alternatively, maybe write it as something else.

Alternatively, maybe write it as 4(2n² + n) + 1. But again, not sure.

Alternatively, maybe check if there's a connection with the original problem.

But since the problem says "simplify the expression", and 8n² + 4n + 1 is a simplified form, unless there's a more elegant expression.

Alternatively, maybe write it as (something). Let me think.

Alternatively, note that 8n² + 4n + 1 = 4(2n² + n) + 1. But again, not helpful.

Alternatively, maybe write it as (something involving (2n + 1)^2).

But since the problem says "simplify", and given that the answer is a quadratic in n, I think 8n² + 4n + 1 is the simplified form.

But let me check for n=1, 8(1)^2 + 4(1) + 1 = 13, which is correct. For n=2, 8(4) + 8 + 1 = 32 + 8 + 1 = 41, which is correct. For n=3, 8(9) + 12 + 1 = 72 + 13 = 85, which is correct.

Therefore, the simplified expression is 8n² + 4n + 1.

But let me check if there's another way to write this. For example, maybe factor out something?

Alternatively, note that 8n² + 4n + 1 = 4(2n² + n) + 1. But not helpful.

Alternatively, maybe write it as (2n + 1)^2 + (2n)^2. Wait, as I did before.

But maybe the problem expects a different form? Let me think again.

Wait, going back to the original problem, the expression was:

[(2^4 + 1/4)(4^4 + 1/4)...(2n)^4 + 1/4] / [(1^4 + 1/4)(3^4 + 1/4)...(2n - 1)^4 + 1/4]

And we found that this simplifies to 8n² + 4n + 1.

Alternatively, maybe there's a way to write this as (something with (2n + 1)^2 or similar).

Alternatively, maybe check if 8n² + 4n + 1 can be written as (something).

Alternatively, maybe think of the original telescoping product.

But since we have already confirmed with multiple examples and the telescoping logic, I think 8n² + 4n + 1 is the correct simplified form.

Therefore, the answer is 8n² + 4n + 1.

But let me check once more for n=1, 2, 3.

n=1: 8(1) + 4(1) + 1 = 13. Correct.

n=2: 8(4) + 4(2) + 1 = 32 + 8 + 1 = 41. Correct.

n=3: 8(9) + 4(3) + 1 = 72 + 12 + 1 = 85. Correct.

Therefore, I think this is the correct answer.

But let me check if there's an alternative expression. For example, maybe (2n + 1)^2 + (2n)^2?

(2n + 1)^2 + (2n)^2 = 4n² + 4n + 1 + 4n² = 8n² + 4n + 1. Yes! So 8n² + 4n + 1 = (2n)^2 + (2n + 1)^2.

But maybe the problem expects this form? Or maybe another form?

Alternatively, maybe write it as (something else).

Alternatively, maybe think of the original expression.

But since the problem says "simplify", and 8n² + 4n + 1 is a simplified polynomial, I think that's acceptable.

Alternatively, maybe factor it as something? But since it doesn't factor over integers, I think that's the simplest.

Therefore, the answer is 8n² + 4n + 1.

But let me check once again the telescoping logic.

We had that D_k = A_{k - 1}, so the product becomes A_n / A_0. Since A_0 = 1, then the product is A_n.

Yes, and A_n = 8n² + 4n + 1.

Therefore, the final answer is 8n² + 4n + 1.

But let me check if there's a different approach.

Alternatively, going back to the original factorization.

We had that each term in the numerator and denominator factors into two quadratics, and after cancellation, we had the product of A_k / D_k, which telescopes to A_n.

Alternatively, maybe there's another way to see this.

But given that multiple examples check out and the telescoping logic is sound, I think this is correct.

Therefore, the simplified expression is 8n² + 4n + 1.

**Final Answer**
\boxed{8n^2 + 4n + 1}

To simplify the expression

$$
\frac{(2^4 + \frac{1}{4})(4^4 + \frac{1}{4}) \dotsm [(2n)^4 + \frac{1}{4}]}{(1^4 + \frac{1}{4})(3^4 + \frac{1}{4}) \dotsm [(2n - 1)^4 + \frac{1}{4}]}
$$

we begin by recognizing a pattern in the terms. Each term in the numerator and denominator is of the form $ x^4 + \frac{1}{4} $, which can be factored using the **Sophie Germain identity**:

$$
x^4 + 4y^4 = (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2)
$$

Setting $ y = \frac{1}{2} $, we get:

$$
x^4 + \frac{1}{4} = (x^2 + x + \frac{1}{2})(x^2 - x + \frac{1}{2})
$$

---

### Step 1: Factor Each Term

Let’s apply this identity to both the numerator and denominator:

- **Numerator terms** (even $ x = 2k $):

$$
(2k)^4 + \frac{1}{4} = (4k^2 + 2k + \frac{1}{2})(4k^2 - 2k + \frac{1}{2})
$$

- **Denominator terms** (odd $ x = 2k - 1 $):

$$
(2k - 1)^4 + \frac{1}{4} = (4k^2 - 2k + \frac{1}{2})(4k^2 - 6k + \frac{5}{2})
$$

---

### Step 2: Write the Full Expression

Now, the full expression becomes:

$$
\frac{\prod_{k=1}^n (4k^2 + 2k + \frac{1}{2})(4k^2 - 2k + \frac{1}{2})}{\prod_{k=1}^n (4k^2 - 2k + \frac{1}{2})(4k^2 - 6k + \frac{5}{2})}
$$

Notice that the second factor in the numerator, $ 4k^2 - 2k + \frac{1}{2} $, is the same as the first factor in the denominator, $ 4k^2 - 2k + \frac{1}{2} $. So, these terms cancel out across the product.

This leaves us with:

$$
\frac{\prod_{k=1}^n (4k^2 + 2k + \frac{1}{2})}{\prod_{k=1}^n (4k^2 - 6k + \frac{5}{2})}
$$

---

### Step 3: Simplify the Remaining Product

Let’s define:

- $ A_k = 4k^2 + 2k + \frac{1}{2} $
- $ D_k = 4k^2 - 6k + \frac{5}{2} $

Then the expression becomes:

$$
\prod_{k=1}^n \frac{A_k}{D_k}
$$

Now, observe that:

$$
D_k = A_{k-1}
$$

This is because:

$$
A_{k-1} = 4(k-1)^2 + 2(k-1) + \frac{1}{2} = 4k^2 - 8k + 4 + 2k - 2 + \frac{1}{2} = 4k^2 - 6k + \frac{5}{2}
$$

So, the product becomes:

$$
\prod_{k=1}^n \frac{A_k}{A_{k-1}} = \frac{A_n}{A_0}
$$

Since $ A_0 = 4(0)^2 + 2(0) + \frac{1}{2} = \frac{1}{2} $, we have:

$$
\frac{A_n}{A_0} = \frac{A_n}{\frac{1}{2}} = 2A_n
$$

But wait! Actually, we had:

$$
\prod_{k=1}^n \frac{A_k}{D_k} = \prod_{k=1}^n \frac{A_k}{A_{k-1}} = \frac{A_n}{A_0}
$$

So the final expression is:

$$
\frac{A_n}{A_0} = \frac{8n^2 + 4n + 1}{1} = 8n^2 + 4n + 1
$$

---

### Final Answer

$$
\boxed{8n^2 + 4n + 1}
$$

Solution:
Let
\[f(m) = m^4 + \frac{1}{4} = \frac{4m^4 + 1}{4}.\]We can factor this with a little give and take:
\begin{align*}
f(m) &= \frac{4m^4 + 1}{4} \\
&= \frac{4m^4 + 4m^2 + 1 - 4m^2}{4} \\
&= \frac{(2m^2 + 1)^2 - (2m)^2}{4} \\
&= \frac{(2m^2 + 2m + 1)(2m^2 - 2m + 1)}{4}.
\end{align*}Now, let $g(m) = 2m^2 + 2m + 1.$  Then
\[g(m - 1) = 2(m - 1)^2 + 2(m - 1) + 1 = 2m^2 - 2m + 1.\]Hence,
\[f(m) = \frac{g(m) g(m - 1)}{4}.\]Therefore,
\begin{align*}
\frac{(2^4 + \frac{1}{4})(4^4 + \frac{1}{4}) \dotsm [(2n)^4 + \frac{1}{4}]}{(1^4 + \frac{1}{4})(3^4 + \frac{1}{4}) \dotsm [(2n - 1)^4 + \frac{1}{4}]} &= \frac{f(2) f(4) \dotsm f(2n)}{f(1) f(3) \dotsm f(2n - 1)} \\
&= \frac{\frac{g(2) g(1)}{4} \cdot \frac{g(4) g(3)}{4} \dotsm \frac{g(2n) g(2n - 1)}{4}}{\frac{g(1) g(0)}{4} \cdot \frac{g(3) g(2)}{4} \dotsm \frac{g(2n - 1) g(2n - 2)}{4}} \\
&= \frac{g(2n)}{g(0)} \\
&= 2(2n)^2 + 2(2n) + 1 \\
&= \boxed{8n^2 + 4n + 1}.
\end{align*}


Explanation:
Okay, so I need to find the probability that the reciprocal of a randomly selected positive odd integer less than 2010 gives a terminating decimal. Then express that probability as a reduced fraction a/b and find a + b. Alright, let me start by recalling what makes a decimal terminating.

From what I remember, a decimal expansion of a fraction terminates if and only if the denominator, after simplifying the fraction to its lowest terms, has no prime factors other than 2 and 5. So, for example, 1/2 = 0.5, which terminates, and 1/3 = 0.333..., which doesn't. So, if I have a fraction 1/n, then 1/n will have a terminating decimal if and only if n is of the form 2^k * 5^m where k and m are non-negative integers. But wait, the question is about reciprocals of odd integers. So since the integers we are considering are odd, they are not divisible by 2. Therefore, for 1/n to have a terminating decimal, n must be of the form 5^m, right? Because the denominator can't have any prime factors other than 2 and 5, but since n is odd, it can't have 2 as a factor. Therefore, n must be a power of 5. Is that correct?

Let me check. Suppose n is an odd integer. Then, if n is a power of 5, say 5^m, then 1/n will terminate. If n has any other prime factors besides 5, then 1/n will not terminate. For example, 1/15 = 0.0666..., which doesn't terminate. So yes, that seems right. Therefore, the problem reduces to finding how many positive odd integers less than 2010 are powers of 5, and then dividing that by the total number of positive odd integers less than 2010. Then, simplify that fraction and find a + b.

Okay, so first step: find the total number of positive odd integers less than 2010. Since we're dealing with positive integers less than 2010, that's from 1 to 2009 inclusive. The number of odd numbers in this range is equal to the ceiling of 2009/2. Wait, but since 2009 is odd, the number of odd numbers is (2009 + 1)/2 = 2010/2 = 1005. Wait, let me check that again. The number of odd numbers between 1 and N inclusive is floor((N + 1)/2). If N is odd, then (N + 1)/2 is an integer. For example, N = 5: numbers 1, 3, 5: 3 numbers. (5 + 1)/2 = 3. So yes, for N = 2009, which is odd, the number of odd numbers is (2009 + 1)/2 = 2010/2 = 1005. So total number of possible integers is 1005. Got that.

Now, the next part is to find how many of these odd integers are powers of 5. So, we need to find all integers n such that n = 5^k, where n < 2010 and n is odd. Since 5^k is always odd, we don't have to worry about that. So, we need to find all exponents k such that 5^k < 2010.

Let me compute the powers of 5:

5^1 = 5

5^2 = 25

5^3 = 125

5^4 = 625

5^5 = 3125

Wait, 5^5 is 3125, which is greater than 2010. Therefore, the exponents k can be 1, 2, 3, 4. Therefore, there are 4 such numbers: 5, 25, 125, 625. Let me check:

5^1 = 5 < 2010

5^2 = 25 < 2010

5^3 = 125 < 2010

5^4 = 625 < 2010

5^5 = 3125 > 2010. So yes, only up to k=4. Therefore, there are 4 numbers. Therefore, the probability is 4 / 1005. Wait, but we need to check if 4 and 1005 are coprime. Let me check.

First, factorize 1005. 1005 divided by 5 is 201. Then 201 is 3*67. Therefore, 1005 = 5 * 3 * 67. The numerator is 4. The prime factors of 4 are 2^2. Since 4 and 1005 share no common prime factors, they are coprime. Therefore, the fraction is already reduced, so a = 4, b = 1005, so a + b = 4 + 1005 = 1009. Wait, but hold on. Is that correct? Let me check again.

Wait, but maybe I made a mistake here. Let me think again. The question says "positive odd integer less than 2010". So numbers from 1 to 2009 inclusive. But 5^k must be less than 2010. So 5^4 is 625, 5^5 is 3125. So 5^4 is 625, which is less than 2010. So that's four numbers: 5, 25, 125, 625. So that's four numbers. Therefore, the probability is 4 / 1005. But let me check if there are more numbers. Wait, is there another power of 5? For example, 5^0 is 1. Is 1 considered? Wait, the problem says "positive odd integer less than 2010". So 1 is included. But 1 is 5^0. However, does 1 count? Let me check.

If n = 1, then 1/1 = 1.0, which is a terminating decimal. So yes, 1 is a valid number. Therefore, 5^0 = 1 is also a power of 5. Therefore, maybe I missed that. So, the exponents k can be 0, 1, 2, 3, 4. Let me check 5^0 = 1, which is less than 2010. So that's another one. Therefore, there are 5 numbers: 1, 5, 25, 125, 625. Therefore, the count is 5 instead of 4. Therefore, the probability is 5 / 1005. Then, simplifying that fraction. Let me check.

So, 5 / 1005. Divide numerator and denominator by 5: 1 / 201. Therefore, a = 1, b = 201. Then a + b = 202. Wait, but why did I not consider 1 before? Because the problem says "positive odd integer less than 2010". So 1 is included. But when I thought about the reciprocal of 1, which is 1.0, which is terminating. Therefore, 1 should be included. Therefore, my initial count was wrong. So, there are 5 numbers: 1, 5, 25, 125, 625. Therefore, the probability is 5 / 1005. Then simplifying, divide numerator and denominator by 5: 1 / 201. Therefore, a = 1, b = 201. Then a + b = 202. But let me check again if there are more numbers. Let me check 5^5 is 3125, which is more than 2010, so no. So 5^0 is 1, 5^1 is 5, 5^2 is 25, 5^3 is 125, 5^4 is 625. So that's five numbers. Therefore, the count is 5. Therefore, probability is 5 / 1005 = 1 / 201. Therefore, a + b = 1 + 201 = 202. But let me check again if 1 is considered. The problem says "positive odd integer less than 2010". So 1 is a positive odd integer less than 2010. Therefore, yes, it should be included. Therefore, the answer is 202? Wait, but let me check again.

Wait, but maybe there's a mistake here. Let me think again. The original reasoning was that for 1/n to have a terminating decimal, n must be of the form 2^k * 5^m. However, since n is odd, it cannot have factors of 2, so m must be such that n is 5^m. However, 1 is 5^0, so that's allowed. Therefore, n can be 5^0, 5^1, 5^2, etc. So, in that case, the numbers are 1, 5, 25, 125, 625. So five numbers. Therefore, the count is 5. Therefore, probability is 5 / 1005. Simplify that fraction. Let me check if 5 divides into 1005. 1005 divided by 5 is 201. So yes, 5 / 1005 = 1 / 201. Therefore, a = 1, b = 201. Therefore, a + b = 202. But let me check if there are other numbers. For example, numbers that are products of 5s and 1s? Wait, but since n is odd, and we need n to be of the form 5^m. Because if n had any other prime factors, then 1/n would not terminate. Therefore, only numbers that are powers of 5. Therefore, only 1, 5, 25, 125, 625. Therefore, five numbers. Therefore, the answer is 202. But let me check again if 1 is considered. Let me check the problem statement again: "the reciprocal of a randomly selected positive odd integer less than 2010 gives a terminating decimal". So reciprocal of 1 is 1.0, which is terminating. So yes, 1 is included. Therefore, the count is 5. Therefore, probability is 5 / 1005 = 1 / 201. Therefore, answer is 1 + 201 = 202. But wait, let me check if there are more numbers. For example, numbers like 5^0 = 1, 5^1 = 5, 5^2 = 25, 5^3 = 125, 5^4 = 625, 5^5 = 3125. 3125 is more than 2010, so no. Therefore, only five numbers. Therefore, 5 numbers. Therefore, 5 / 1005. Simplify. 5 divides into 1005, as 1005 / 5 = 201. So 5 / 1005 = 1 / 201. Therefore, a = 1, b = 201. Therefore, a + b = 202. Therefore, the answer is 202. But let me check again if there's a mistake here. Maybe I need to check if there are other numbers which are not powers of 5 but still have only 5 as prime factors? Wait, but if n is an odd integer, then n can't have 2 as a factor. So if n is to have only 5 as prime factors, then n must be 5^k. Therefore, there are no other numbers. For example, 25 is 5^2, 125 is 5^3, etc. So, no other numbers. Therefore, only 1, 5, 25, 125, 625. Therefore, five numbers. Therefore, the answer is 202. Wait, but let me check with another approach. Let me think again.

Another way: The probability is equal to the number of odd integers less than 2010 which are of the form 5^k divided by the total number of odd integers less than 2010. So, total number of odd integers less than 2010 is 1005 as before. Number of such n is 5. Therefore, 5 / 1005. Simplify. 5 divides into 1005, so divide numerator and denominator by 5: 1 / 201. Therefore, 1 + 201 = 202. Therefore, answer is 202. But let me check if 1 is considered. Suppose someone argues that 1 is not considered because it's trivial? But the problem says "positive odd integer", so 1 is included. Therefore, I think that is correct. However, let me check with an example. Let me take a smaller number instead of 2010. Let's say, instead of less than 2010, let's say less than 10. Then, the odd numbers less than 10 are 1, 3, 5, 7, 9. The reciprocals: 1/1 = 1.0, terminating; 1/3 = 0.333..., non-terminating; 1/5 = 0.2, terminating; 1/7 ≈ 0.1428..., non-terminating; 1/9 ≈ 0.111..., non-terminating. Therefore, there are two numbers: 1 and 5. Therefore, probability is 2 / 5. According to our formula, the number of powers of 5 less than 10: 5^0 = 1, 5^1 = 5, 5^2 = 25 which is more than 10. Therefore, two numbers. Total odd numbers less than 10 is 5. Therefore, 2 / 5. Which is correct. Therefore, in this case, the answer would be 2 / 5, which reduces to 2 / 5, so a + b = 7. So, in this case, 1 is included. Therefore, seems like 1 is included. Therefore, in our original problem, 1 is included. Therefore, answer is 202. Therefore, I think that is correct. Therefore, the answer is 202. Wait, but let me check another example. Let me take numbers less than 25. Then, odd numbers less than 25: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23. Total of 12 numbers. The numbers which are powers of 5: 1, 5, 25. But 25 is not less than 25, so only 1 and 5. Therefore, 2 numbers. Therefore, probability is 2 / 12 = 1 / 6. According to our formula, number of powers of 5 less than 25: 5^0 = 1, 5^1 = 5, 5^2 = 25 which is not less than 25. Therefore, two numbers. Therefore, 2 / 12 = 1 / 6. Which is correct. Therefore, in this case, 1 is included. Therefore, seems like 1 is included. Therefore, in our original problem, 1 is included. Therefore, answer is 202. Therefore, I think that is correct. Therefore, the answer is 202. However, let me check once again if there are other numbers. For example, numbers like 5^k multiplied by 1? But 1 is already included. Wait, no. If n is 5^k, then n is already a power of 5. So, there are no other numbers. Therefore, only 1, 5, 25, 125, 625. Therefore, five numbers. Therefore, 5 / 1005 = 1 / 201. Therefore, answer is 202. Therefore, I think that is correct. Therefore, the answer is 202. Wait, but let me check if 1 is considered. Suppose someone says that 1 is not considered because it's not a "positive odd integer greater than 1"? But the problem says "positive odd integer less than 2010", so 1 is included. Therefore, I think that is correct. Therefore, I think the answer is 202. Therefore, I think the answer is \boxed{202}.

**Final Answer**
\boxed{202}

To solve the problem, we are asked to find the probability that the reciprocal of a randomly selected **positive odd integer less than 2010** gives a **terminating decimal**, and then express this probability as a reduced fraction $ \frac{a}{b} $, where $ a $ and $ b $ are relatively prime positive integers. Finally, we are to compute $ a + b $.

---

### Step 1: Understanding Terminating Decimals

A fraction $ \frac{1}{n} $ has a **terminating decimal** if and only if the **denominator**, when the fraction is in lowest terms, has **no prime factors other than 2 and 5**. That is, $ n $ must be of the form $ 2^k \cdot 5^m $, where $ k, m \geq 0 $.

However, in this problem, we are only considering **odd integers**, so $ n $ cannot be divisible by 2. Therefore, the only allowed prime factor is **5**. Thus, for $ \frac{1}{n} $ to have a terminating decimal, $ n $ must be a **power of 5**.

---

### Step 2: Counting the Total Number of Positive Odd Integers Less Than 2010

We are selecting from all **positive odd integers less than 2010**. These are the numbers:

$$
1, 3, 5, 7, \ldots, 2009
$$

This is an arithmetic sequence with first term 1, last term 2009, and common difference 2. The number of terms is:

$$
\frac{2009 - 1}{2} + 1 = \frac{2008}{2} + 1 = 1004 + 1 = 1005
$$

So, there are **1005** such numbers.

---

### Step 3: Counting the Number of Powers of 5 Less Than 2010

We now count how many **odd integers less than 2010** are powers of 5. Since all powers of 5 are odd, we only need to find the number of powers of 5 less than 2010.

Compute successive powers of 5:

- $ 5^0 = 1 $
- $ 5^1 = 5 $
- $ 5^2 = 25 $
- $ 5^3 = 125 $
- $ 5^4 = 625 $
- $ 5^5 = 3125 $ (which is **greater than 2010**)

So, the valid powers of 5 less than 2010 are:

$$
1, 5, 25, 125, 625
$$

That is, **5** such numbers.

---

### Step 4: Calculating the Probability

The probability that a randomly selected odd integer less than 2010 has a reciprocal that is a terminating decimal is:

$$
\frac{5}{1005}
$$

Now, simplify the fraction:

$$
\frac{5}{1005} = \frac{1}{201}
$$

So, $ a = 1 $, $ b = 201 $, and $ a + b = 1 + 201 = \boxed{202} $

---

### Final Answer

$$
\boxed{202}
$$

[Assistant]
Explanation:
Okay, so I need to find the probability that the reciprocal of a randomly selected positive odd integer less than 2010 gives a terminating decimal. Then express that probability as a reduced fraction a/b and find a + b. Alright, let me start by recalling what makes a decimal terminating.

From what I remember, a decimal expansion of a fraction terminates if and only if the denominator, after simplifying the fraction to its lowest terms, has no prime factors other than 2 and 5. So, for example, 1/2 = 0.5, which terminates, and 1/3 = 0.333..., which doesn't. So, if I have a fraction 1/n, then 1/n will have a terminating decimal if and only if n is of the form 2^k * 5^m where k and m are non-negative integers. But wait, the question is about reciprocals of odd integers. So since the integers we are considering are odd, they are not divisible by 2. Therefore, for 1/n to have a terminating decimal, n must be of the form 5^m, right? Because the denominator can't have any prime factors other than 2 and 5, but since n is odd, it can't have 2 as a factor. Therefore, n must be a power of 5. Is that correct?

Let me check. Suppose n is an odd integer. Then, if n is a power of 5, say 5^m, then 1/n will terminate. If n has any other prime factors besides 5, then 1/n will not terminate. For example, 1/15 = 0.0666..., which doesn't terminate. So yes, that seems right. Therefore, the problem reduces to finding how many positive odd integers less than 2010 are powers of 5, and then dividing that by the total number of positive odd integers less than 2010. Then, simplify that fraction and find a + b.

Okay, so first step: find the total number of positive odd integers less than 2010. Since we're dealing with positive integers less than 2010, that's from 1 to 2009 inclusive. The number of odd numbers in this range is equal to the ceiling of 2009/2. Wait, but since 2009 is odd, the number of odd numbers is (2009 + 1)/2 = 2010/2 = 1005. Wait, let me check that again. The number of odd numbers between 1 and N inclusive is floor((N + 1)/2). If N is odd, then (N + 1)/2 is an integer. For example, N = 5: numbers 1, 3, 5: 3 numbers. (5 + 1)/2 = 3. So yes, for N = 2009, which is odd, the number of odd numbers is (2009 + 1)/2 = 2010/2 = 1005. So total number of possible integers is 1005. Got that.

Now, the next part is to find how many of these odd integers are powers of 5. So, we need to find all integers n such that n = 5^k, where n < 2010 and n is odd. Since 5^k is always odd, we don't have to worry about that. So, we need to find all exponents k such that 5^k < 2010.

Let me compute the powers of 5:

5^1 = 5

5^2 = 25

5^3 = 125

5^4 = 625

5^5 = 3125

Wait, 5^5 is 3125, which is greater than 2010. Therefore, the exponents k can be 1, 2, 3, 4. Therefore, there are 4 such numbers: 5, 25, 125, 625. Let me check:

5^1 = 5 < 2010

5^2 = 25 < 2010

5^3 = 125 < 2010

5^4 = 625 < 2010

5^5 = 3125 > 2010. So yes, only up to k=4. Therefore, there are 4 numbers. Therefore, the probability is 4 / 1005. Wait, but we need to check if 4 and 1005 are coprime. Let me check.

First, factorize 1005. 1005 divided by 5 is 201. Then 201 is 3*67. Therefore, 1005 = 5 * 3 * 67. The numerator is 4. The prime factors of 4 are 2^2. Since 4 and 1005 share no common prime factors, they are coprime. Therefore, the fraction is already reduced, so a = 4, b = 1005, so a + b = 4 + 1005 = 1009. Wait, but hold on. Is that correct? Let me check again.

Wait, but maybe I made a mistake here. Let me think again. The question says "positive odd integer less than 2010". So numbers from 1 to 2009 inclusive. But 5^k must be less than 2010. So 5^4 is 625, 5^5 is 3125. So 5^4 is 625, which is less than 2010. So that's four numbers: 5, 25, 125, 625. So that's four numbers. Therefore, the probability is 4 / 1005. But let me check if there are more numbers. Wait, is there another power of 5? For example, 5^0 is 1. Is 1 considered? Wait, the problem says "positive odd integer less than 2010". So 1 is included. But 1 is 5^0. However, does 1 count? Let me check.

If n = 1, then 1/1 = 1.0, which is a terminating decimal. So yes, 1 is a valid number. Therefore, 5^0 = 1 is also a power of 5. Therefore, maybe I missed that. So, the exponents k can be 0, 1, 2, 3, 4. Let me check 5^0 = 1, which is less than 2010. So that's another one. Therefore, there are 5 numbers: 1, 5, 25, 125, 625. Therefore, the count is 5 instead of 4. Therefore, the probability is 5 / 1005. Then, simplifying that fraction. Let me check.

So, 5 / 1005. Divide numerator and denominator by 5: 1 / 201. Therefore, a = 1, b = 201. Then a + b = 202. Wait, but why did I not consider 1 before? Because the problem says "positive odd integer less than 2010". So 1 is included. But when I thought about the reciprocal of 1, which is 1.0, which is terminating. Therefore, 1 should be included. Therefore, my initial count was wrong. So, there are 5 numbers: 1, 5, 25, 125, 625. Therefore, the probability is 5 / 1005. Then simplifying, divide numerator and denominator by 5: 1 / 201. Therefore, a = 1, b = 201. Then a + b = 202. But let me check again if there are more numbers. Let me check 5^5 is 3125, which is more than 2010, so no. So 5^0 is 1, 5^1 is 5, 5^2 is 25, 5^3 is 125, 5^4 is 625. So that's five numbers. Therefore, the count is 5. Therefore, probability is 5 / 1005 = 1 / 201. Therefore, a + b = 1 + 201 = 202. But let me check again if 1 is considered. The problem says "positive odd integer less than 2010". So 1 is a positive odd integer less than 2010. Therefore, yes, it should be included. Therefore, the answer is 202? Wait, but let me check again.

Wait, but maybe there's a mistake here. Let me think again. The original reasoning was that for 1/n to have a terminating decimal, n must be of the form 2^k * 5^m. However, since n is odd, it cannot have factors of 2, so m must be such that n is 5^m. However, 1 is 5^0, so that's allowed. Therefore, n can be 5^0, 5^1, 5^2, etc. So, in that case, the numbers are 1, 5, 25, 125, 625. So five numbers. Therefore, the count is 5. Therefore, probability is 5 / 1005. Simplify that fraction. Let me check if 5 divides into 1005. 1005 divided by 5 is 201. So yes, 5 / 1005 = 1 / 201. Therefore, a = 1, b = 201. Therefore, a + b = 202. But let me check if there are other numbers. For example, numbers that are products of 5s and 1s? Wait, but since n is odd, and we need n to be of the form 5^m. Because if n had any other prime factors, then 1/n would not terminate. Therefore, only numbers that are powers of 5. Therefore, only 1, 5, 25, 125, 625. Therefore, five numbers. Therefore, the answer is 202. But let me check again if 1 is considered. Let me check the problem statement again: "the reciprocal of a randomly selected positive odd integer less than 2010 gives a terminating decimal". So reciprocal of 1 is 1.0, which is terminating. So yes, 1 is included. Therefore, the count is 5. Therefore, probability is 5 / 1005 = 1 / 201. Therefore, answer is 1 + 201 = 202. But wait, let me check if there are more numbers. For example, numbers like 5^0 = 1, 5^1 = 5, 5^2 = 25, 5^3 = 125, 5^4 = 625, 5^5 = 3125. 3125 is more than 2010, so no. Therefore, only five numbers. Therefore, 5 numbers. Therefore, 5 / 1005. Simplify. 5 divides into 1005, as 1005 / 5 = 201. So 5 / 1005 = 1 / 201. Therefore, a = 1, b = 201. Therefore, a + b = 202. Therefore, the answer is 202. But let me check again if there's a mistake here. Maybe I need to check if there are other numbers which are not powers of 5 but still have only 5 as prime factors? Wait, but if n is an odd integer, then n can't have 2 as a factor. So if n is to have only 5 as prime factors, then n must be 5^k. Therefore, there are no other numbers. For example, 25 is 5^2, 125 is 5^3, etc. So, no other numbers. Therefore, only 1, 5, 25, 125, 625. Therefore, five numbers. Therefore, the answer is 202. Wait, but let me check with another approach. Let me think again.

Another way: The probability is equal to the number of odd integers less than 2010 which are of the form 5^k divided by the total number of odd integers less than 2010. So, total number of odd integers less than 2010 is 1005 as before. Number of such n is 5. Therefore, 5 / 1005. Simplify. 5 divides into 1005, so divide numerator and denominator by 5: 1 / 201. Therefore, 1 + 201 = 202. Therefore, answer is 202. But let me check if 1 is considered. Suppose someone argues that 1 is not considered because it's trivial? But the problem says "positive odd integer", so 1 is included. Therefore, I think that is correct. However, let me check with an example. Let me take a smaller number instead of 2010. Let's say, instead of less than 2010, let's say less than 10. Then, the odd numbers less than 10 are 1, 3, 5, 7, 9. The reciprocals: 1/1 = 1.0, terminating; 1/3 = 0.333..., non-terminating; 1/5 = 0.2, terminating; 1/7 ≈ 0.1428..., non-terminating; 1/9 ≈ 0.111..., non-terminating. Therefore, there are two numbers: 1 and 5. Therefore, probability is 2 / 5. According to our formula, the number of powers of 5 less than 10: 5^0 = 1, 5^1 = 5, 5^2 = 25 which is more than 10. Therefore, two numbers. Total odd numbers less than 10 is 5. Therefore, 2 / 5. Which is correct. Therefore, in this case, the answer would be 2 / 5, which reduces to 2 / 5, so a + b = 7. So, in this case, 1 is included. Therefore, seems like 1 is included. Therefore, in our original problem, 1 is included. Therefore, answer is 202. Therefore, I think that is correct. Therefore, the answer is 202. Wait, but let me check another example. Let me take numbers less than 25. Then, odd numbers less than 25: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23. Total of 12 numbers. The numbers which are powers of 5: 1, 5, 25. But 25 is not less than 25, so only 1 and 5. Therefore, 2 numbers. Therefore, probability is 2 / 12 = 1 / 6. According to our formula, number of powers of 5 less than 25: 5^0 = 1, 5^1 = 5, 5^2 = 25 which is not less than 25. Therefore, two numbers. Therefore, 2 / 12 = 1 / 6. Which is correct. Therefore, in this case, 1 is included. Therefore, seems like 1 is included. Therefore, in our original problem, 1 is included. Therefore, answer is 202. Therefore, I think that is correct. Therefore, the answer is 202. However, let me check once again if there are other numbers. For example, numbers like 5^k multiplied by 1? But 1 is already included. Wait, no. If n is 5^k, then n is already a power of 5. So, there are no other numbers. Therefore, only 1, 5, 25, 125, 625. Therefore, five numbers. Therefore, 5 / 1005 = 1 / 201. Therefore, answer is 202. Therefore, I think that is correct. Therefore, the answer is 202. Wait, but let me check if 1 is considered. Suppose someone says that 1 is not considered because it's not a "positive odd integer greater than 1"? But the problem says "positive odd integer less than 2010", so 1 is included. Therefore, I think that is correct. Therefore, I think the answer is 202. Therefore, I think the answer is \boxed{202}.

**Final Answer**
\boxed{202}

To solve the problem, we are asked to find the probability that the reciprocal of a randomly selected **positive odd integer less than 2010** gives a **terminating decimal**, and then express this probability as a reduced fraction $ \frac{a}{b} $, where $ a $ and $ b $ are relatively prime positive integers. Finally, we are to compute $ a + b $.

---

### Step 1: Understanding Terminating Decimals

A fraction $ \frac{1}{n} $ has a **terminating decimal** if and only if the **denominator**, when the fraction is in lowest terms, has **no prime factors other than 2 and 5**. That is, $ n $ must be of the form $ 2^k \cdot 5^m $, where $ k, m \geq 0 $.

However, in this problem, we are only considering **odd integers**, so $ n $ cannot be divisible by 2. Therefore, the only allowed prime factor is **5**. Thus, for $ \frac{1}{n} $ to have a terminating decimal, $ n $ must be a **power of 5**.

---

### Step 2: Counting the Total Number of Positive Odd Integers Less Than 2010

We are selecting from all **positive odd integers less than 2010**. These are the numbers:

$$
1, 3, 5, 7, \ldots, 2009
$$

This is an arithmetic sequence with first term 1, last term 2009, and common difference 2. The number of terms is:

$$
\frac{2009 - 1}{2} + 1 = \frac{2008}{2} + 1 = 1004 + 1 = 1005
$$

So, there are **1005** such numbers.

---

### Step 3: Counting the Number of Powers of 5 Less Than 2010

We now count how many **odd integers less than 2010** are powers of 5. Since all powers of 5 are odd, we only need to find the number of powers of 5 less than 2010.

Compute successive powers of 5:

- $ 5^0 = 1 $
- $ 5^1 = 5 $
- $ 5^2 = 25 $
- $ 5^3 = 125 $
- $ 5^4 = 625 $
- $ 5^5 = 3125 $ (which is **greater than 2010**)

So, the valid powers of 5 less than 2010 are:

$$
1, 5, 25, 125, 625
$$

That is, **5** such numbers.

---

### Step 4: Calculating the Probability

The probability that a randomly selected odd integer less than 2010 has a reciprocal that is a terminating decimal is:

$$
\frac{5}{1005}
$$

Now, simplify the fraction:

$$
\frac{5}{1005} = \frac{1}{201}
$$

So, $ a = 1 $, $ b = 201 $, and $ a + b = 1 + 201 = \boxed{202} $

---

### Final Answer

$$
\boxed{202}
$$

Solution:
There are 2009 positive integers less than 2010, of which 1005 are odd. If $\frac{1}{n}$ is equal to a terminating decimal, then $n$ can only be divisible by 2 and 5. However, since we have the added restriction that $n$ is odd, $n$ must be a power of 5. There are five powers of 5 less than 2010. \begin{align*}
5^0 &= 1 \\
5^1 &= 5 \\
5^2 &= 25 \\
5^3 &= 125 \\
5^4 &= 625
\end{align*} Note that $5^5 = 3125$. Since there are five odd integers that satisfy our desired condition, the desired probability is $\frac{5}{1005} = \frac{1}{201}$. This is in simplest terms, so our answer is $1+201 = \boxed{202}$.


Explanation:
Okay, so I need to find the original price of a shirt when the sale price is $14.40, and that sale price is 60% off the original price. Hmm, let me think.

First, let me make sure I understand what "60% off" means. If something is 60% off, that means the customer is paying 40% of the original price, right? Because 100% - 60% = 40%. So the sale price is 40% of the original price.

So, if I let the original price be some amount, let's call it "P", then 40% of P equals $14.40.

Mathematically, that would be 0.40 * P = 14.40.

So, to find P, I need to divide both sides of the equation by 0.40.

Let me write that down:

P = 14.40 / 0.40

Hmm, let me calculate that. Dividing by 0.40 is the same as multiplying by 2.5, right? Because 1 divided by 0.4 is 2.5. Let me check that. 0.4 times 2.5 is 1.0, yes. So 14.40 divided by 0.40 is 14.40 * 2.5.

Let me compute 14.40 * 2.5. Breaking it down, 14.40 * 2 is 28.80, and 14.40 * 0.5 is 7.20. Adding those together: 28.80 + 7.20 = 36.00.

So, the original price was $36.00? Let me verify that. If the original price is $36, then 60% off would be 0.60 * 36 = 21.60. Subtracting that discount from the original price: 36 - 21.60 = 14.40. Yep, that matches the sale price given.

Wait, but let me make sure I didn't mix up anything. Sometimes when people say "discount," they might mean different things, but in this case, it's straightforward. 60% off means the discount is 60%, so the sale price is original minus 60% of original, which is 40% of original. So my calculation seems right.

Alternatively, if I use the equation again: 0.40 * P = 14.40. So P = 14.40 / 0.40. Let me do this division another way.

Dividing 14.40 by 0.40. To make it easier, I can multiply numerator and denominator by 100 to eliminate decimals. That gives me 1440 / 40.

Calculating 1440 divided by 40. Well, 40 goes into 1440 how many times? 40 * 30 = 1200. 1440 - 1200 = 240. 40 * 6 = 240. So total is 30 + 6 = 36. Yep, same answer.

So that confirms it. The original price was $36.

Wait, just to check once more. If the original price is $36, then 60% of that is 0.6 * 36. Let me compute that. 0.6 * 30 = 18, and 0.6 * 6 = 3.6, so total discount is 18 + 3.6 = 21.6. Subtract that from 36: 36 - 21.6 = 14.4. Perfect, that's the sale price.

So I think that's solid. The original price was $36.

But let me think if there's another way someone might approach this problem and maybe get confused. For example, someone might think that 60% off means the sale price is 60% of the original price, but that would be incorrect. Because "off" means subtracting from the original. So 60% off is subtracting 60% of the original, so the remaining is 40%.

Alternatively, if someone thought that 60% off is 60% of the original price, then they would do 14.40 = 0.60 * P, leading to P = 14.40 / 0.60 = 24. But that would be wrong because that would mean the sale price is 60% of original, but the problem says it's 60% off.

So, it's important to distinguish between "discount percentage" and "remaining percentage." So, if something is 60% off, the remaining is 40%, so the sale price is 40% of original.

Therefore, my answer of $36 seems correct.

Another way to think about it: if 60% is taken off, then the sale price is 40% of original. So, $14.40 is 40% of original. So, 10% would be 14.40 divided by 4, which is 3.60. Then 100% would be 3.60 * 10 = 36.00. Yep, same result.

So, multiple methods all leading to $36.

Therefore, I'm confident that the original price was $36.

**Final Answer**
The original price of the shirt was \boxed{36} dollars.

To determine the original price of the shirt, we start by understanding the relationship between the sale price and the original price.

---

### Step 1: Understand the Discount

The problem states that the sale price is **60% off** the original price. This means the customer is paying **40%** of the original price. In mathematical terms:

$$
\text{Sale Price} = 40\% \text{ of Original Price}
$$

Let the original price be $ P $. Then:

$$
0.40 \times P = 14.40
$$

---

### Step 2: Solve for the Original Price

To find $ P $, divide both sides of the equation by 0.40:

$$
P = \frac{14.40}{0.40}
$$

We can simplify this division by multiplying numerator and denominator by 100 to eliminate decimals:

$$
P = \frac{1440}{40} = 36
$$

---

### Step 3: Verify the Result

To confirm, calculate 60% of $36:

$$
0.60 \times 36 = 21.60
$$

Subtract this discount from the original price:

$$
36 - 21.60 = 14.40
$$

This matches the given sale price, confirming our calculation is correct.

---

### Final Answer

$$
\boxed{36}
$$

[Assistant]
Explanation:
Okay, so I need to find the original price of a shirt when the sale price is $14.40, and that sale price is 60% off the original price. Hmm, let me think.

First, let me make sure I understand what "60% off" means. If something is 60% off, that means the customer is paying 40% of the original price, right? Because 100% - 60% = 40%. So the sale price is 40% of the original price.

So, if I let the original price be some amount, let's call it "P", then 40% of P equals $14.40.

Mathematically, that would be 0.40 * P = 14.40.

So, to find P, I need to divide both sides of the equation by 0.40.

Let me write that down:

P = 14.40 / 0.40

Hmm, let me calculate that. Dividing by 0.40 is the same as multiplying by 2.5, right? Because 1 divided by 0.4 is 2.5. Let me check that. 0.4 times 2.5 is 1.0, yes. So 14.40 divided by 0.40 is 14.40 * 2.5.

Let me compute 14.40 * 2.5. Breaking it down, 14.40 * 2 is 28.80, and 14.40 * 0.5 is 7.20. Adding those together: 28.80 + 7.20 = 36.00.

So, the original price was $36.00? Let me verify that. If the original price is $36, then 60% off would be 0.60 * 36 = 21.60. Subtracting that discount from the original price: 36 - 21.60 = 14.40. Yep, that matches the sale price given.

Wait, but let me make sure I didn't mix up anything. Sometimes when people say "discount," they might mean different things, but in this case, it's straightforward. 60% off means the discount is 60%, so the sale price is original minus 60% of original, which is 40% of original. So my calculation seems right.

Alternatively, if I use the equation again: 0.40 * P = 14.40. So P = 14.40 / 0.40. Let me do this division another way.

Dividing 14.40 by 0.40. To make it easier, I can multiply numerator and denominator by 100 to eliminate decimals. That gives me 1440 / 40.

Calculating 1440 divided by 40. Well, 40 goes into 1440 how many times? 40 * 30 = 1200. 1440 - 1200 = 240. 40 * 6 = 240. So total is 30 + 6 = 36. Yep, same answer.

So that confirms it. The original price was $36.

Wait, just to check once more. If the original price is $36, then 60% of that is 0.6 * 36. Let me compute that. 0.6 * 30 = 18, and 0.6 * 6 = 3.6, so total discount is 18 + 3.6 = 21.6. Subtract that from 36: 36 - 21.6 = 14.4. Perfect, that's the sale price.

So I think that's solid. The original price was $36.

But let me think if there's another way someone might approach this problem and maybe get confused. For example, someone might think that 60% off means the sale price is 60% of the original price, but that would be incorrect. Because "off" means subtracting from the original. So 60% off is subtracting 60% of the original, so the remaining is 40%.

Alternatively, if someone thought that 60% off is 60% of the original price, then they would do 14.40 = 0.60 * P, leading to P = 14.40 / 0.60 = 24. But that would be wrong because that would mean the sale price is 60% of original, but the problem says it's 60% off.

So, it's important to distinguish between "discount percentage" and "remaining percentage." So, if something is 60% off, the remaining is 40%, so the sale price is 40% of original.

Therefore, my answer of $36 seems correct.

Another way to think about it: if 60% is taken off, then the sale price is 40% of original. So, $14.40 is 40% of original. So, 10% would be 14.40 divided by 4, which is 3.60. Then 100% would be 3.60 * 10 = 36.00. Yep, same result.

So, multiple methods all leading to $36.

Therefore, I'm confident that the original price was $36.

**Final Answer**
The original price of the shirt was \boxed{36} dollars.

To determine the original price of the shirt, we start by understanding the relationship between the sale price and the original price.

---

### Step 1: Understand the Discount

The problem states that the sale price is **60% off** the original price. This means the customer is paying **40%** of the original price. In mathematical terms:

$$
\text{Sale Price} = 40\% \text{ of Original Price}
$$

Let the original price be $ P $. Then:

$$
0.40 \times P = 14.40
$$

---

### Step 2: Solve for the Original Price

To find $ P $, divide both sides of the equation by 0.40:

$$
P = \frac{14.40}{0.40}
$$

We can simplify this division by multiplying numerator and denominator by 100 to eliminate decimals:

$$
P = \frac{1440}{40} = 36
$$

---

### Step 3: Verify the Result

To confirm, calculate 60% of $36:

$$
0.60 \times 36 = 21.60
$$

Subtract this discount from the original price:

$$
36 - 21.60 = 14.40
$$

This matches the given sale price, confirming our calculation is correct.

---

### Final Answer

$$
\boxed{36}
$$

Solution:
If the shirt is $60\%$ off, it is currently $.4$ of the original price.  Thus the original price was

$$\frac{\$14.40}{.4}=\boxed{\$36}$$