Explanation:
Okay, so I need to find the value of n such that the sum of the first n terms of the geometric sequence {1/4, 1/8, 1/16, ...} is equal to 255/512. Let me start by recalling the formula for the sum of the first n terms of a geometric sequence.

The formula is S_n = a_1 * (1 - r^n) / (1 - r), where a_1 is the first term, r is the common ratio, and n is the number of terms. Let me verify if I remember this correctly. Yeah, I think that's right. The sum of a geometric series is calculated using that formula when the common ratio r is not equal to 1.

First, let me identify the first term a_1 and the common ratio r from the given sequence. The sequence is {1/4, 1/8, 1/16, ...}. So the first term a_1 is 1/4. To find the common ratio r, I can divide the second term by the first term. Let me do that: (1/8) divided by (1/4) is equal to (1/8) * (4/1) = 4/8 = 1/2. Let me check with the next terms to make sure. The third term is 1/16, so dividing the third term by the second term: (1/16) / (1/8) = (1/16) * (8/1) = 8/16 = 1/2. Yep, so the common ratio r is 1/2.

So now I have a_1 = 1/4, r = 1/2, and the sum S_n is given as 255/512. I need to solve for n. Let me plug these values into the formula:

S_n = (1/4) * (1 - (1/2)^n) / (1 - 1/2)

Let me simplify the denominator first. 1 - 1/2 is 1/2. So the formula becomes:

S_n = (1/4) * (1 - (1/2)^n) / (1/2)

Dividing by 1/2 is the same as multiplying by 2, so:

S_n = (1/4) * 2 * (1 - (1/2)^n) = (1/2) * (1 - (1/2)^n)

So S_n = (1/2) * [1 - (1/2)^n]. And this is equal to 255/512. Let me write that equation:

(1/2) * [1 - (1/2)^n] = 255/512

To solve for n, I can first multiply both sides by 2 to eliminate the 1/2:

1 - (1/2)^n = (255/512) * 2 = 255/256

Wait, let me check that calculation. 255/512 multiplied by 2 is 255/256? Let me compute 255/512 * 2. 2 divided by 512 is 1/256, so 255 * 2 / 512 = 510/512, which simplifies to 255/256. Yes, that's correct. So:

1 - (1/2)^n = 255/256

Now, let me solve for (1/2)^n. Subtract 255/256 from both sides:

1 - 255/256 = (1/2)^n

Calculating the left side: 1 is 256/256, so 256/256 - 255/256 = 1/256. Therefore:

(1/2)^n = 1/256

Now, I need to solve for n in the equation (1/2)^n = 1/256. Let me recall that (1/2)^n is the same as 2^(-n). Similarly, 1/256 is 2^(-8) because 2^8 is 256. Let me verify that: 2^10 is 1024, so 2^8 is 256. Yes, correct. Therefore:

2^(-n) = 2^(-8)

Since the bases are equal, the exponents must be equal. Therefore:

-n = -8 => n = 8

Wait, so n is 8? Let me check if this makes sense. Let me compute the sum of the first 8 terms and see if it equals 255/512.

First, let me compute S_8 using the formula. S_n = (1/2) * [1 - (1/2)^n]. Plugging in n=8:

S_8 = (1/2) * [1 - (1/2)^8] = (1/2) * [1 - 1/256] = (1/2) * (255/256) = 255/512.

Yes! That's exactly the value given in the problem. So n=8 is the correct answer.

But let me just make sure I didn't make any mistakes in my steps. Let me go through again:

1. Identified a_1 = 1/4, r = 1/2. Correct.

2. Applied the formula S_n = a_1*(1 - r^n)/(1 - r). Correct.

3. Plugged in the values: (1/4)*(1 - (1/2)^n)/(1 - 1/2). Correct.

4. Simplified denominator to 1/2, so multiplied by 2: (1/4)*2*(1 - (1/2)^n) = 1/2*(1 - (1/2)^n). Correct.

5. Set equal to 255/512: 1/2*(1 - (1/2)^n) = 255/512. Correct.

6. Multiply both sides by 2: 1 - (1/2)^n = 255/256. Correct.

7. Subtract 255/256 from 1: 1/256. Correct.

8. So (1/2)^n = 1/256. Which is 2^(-n) = 2^(-8). Therefore, n=8. Correct.

9. Checked by plugging back in: S_8 = 255/512. Correct.

Therefore, the answer is n=8.

But just to be thorough, let me compute the sum manually for n=8 and see if it's indeed 255/512. Let's compute each term:

First term: 1/4 = 1/4

Second term: 1/8

Third term: 1/16

Fourth term: 1/32

Fifth term: 1/64

Sixth term: 1/128

Seventh term: 1/256

Eighth term: 1/512

Now, let's add them up:

Let me convert all to denominators of 512 to add them:

1/4 = 128/512

1/8 = 64/512

1/16 = 32/512

1/32 = 16/512

1/64 = 8/512

1/128 = 4/512

1/256 = 2/512

1/512 = 1/512

Now, adding all numerators:

128 + 64 = 192

192 + 32 = 224

224 + 16 = 240

240 + 8 = 248

248 + 4 = 252

252 + 2 = 254

254 + 1 = 255

So total is 255/512. Perfect! That matches the given sum. Therefore, n=8 is indeed the correct answer.

I think that's solid. I don't see any mistakes in my reasoning. So the answer is 8.

**Final Answer**
The value of $ n $ is \boxed{8}.

To solve the problem, we are given a geometric sequence:

$$
\left\{\frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots \right\}
$$

and told that the sum of the first $ n $ terms is:

$$
\frac{255}{512}
$$

We are to find the value of $ n $.

---

### Step 1: Identify the first term and common ratio

From the sequence:

- First term: $ a_1 = \frac{1}{4} $
- Common ratio: $ r = \frac{1/8}{1/4} = \frac{1}{2} $

---

### Step 2: Use the formula for the sum of the first $ n $ terms of a geometric sequence

The formula is:

$$
S_n = a_1 \cdot \frac{1 - r^n}{1 - r}
$$

Substituting the known values:

$$
S_n = \frac{1}{4} \cdot \frac{1 - \left(\frac{1}{2}\right)^n}{1 - \frac{1}{2}} = \frac{1}{4} \cdot \frac{1 - \left(\frac{1}{2}\right)^n}{\frac{1}{2}} = \frac{1}{4} \cdot 2 \cdot \left(1 - \left(\frac{1}{2}\right)^n\right)
$$

Simplify:

$$
S_n = \frac{1}{2} \left(1 - \left(\frac{1}{2}\right)^n\right)
$$

We are told that this equals $ \frac{255}{512} $, so:

$$
\frac{1}{2} \left(1 - \left(\frac{1}{2}\right)^n\right) = \frac{255}{512}
$$

---

### Step 3: Solve for $ n $

Multiply both sides by 2:

$$
1 - \left(\frac{1}{2}\right)^n = \frac{255}{256}
$$

Subtract $ \frac{255}{256} $ from 1:

$$
\left(\frac{1}{2}\right)^n = 1 - \frac{255}{256} = \frac{1}{256}
$$

Now, express both sides as powers of 2:

$$
\left(\frac{1}{2}\right)^n = 2^{-n}, \quad \frac{1}{256} = 2^{-8}
$$

So:

$$
2^{-n} = 2^{-8} \Rightarrow n = 8
$$

---

### Step 4: Verify the result

Let’s compute the sum of the first 8 terms using the formula:

$$
S_8 = \frac{1}{2} \left(1 - \left(\frac{1}{2}\right)^8\right) = \frac{1}{2} \left(1 - \frac{1}{256}\right) = \frac{1}{2} \cdot \frac{255}{256} = \frac{255}{512}
$$

This matches the given sum, confirming that $ n = 8 $ is correct.

---

### Final Answer

$$
\boxed{8}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of n such that the sum of the first n terms of the geometric sequence {1/4, 1/8, 1/16, ...} is equal to 255/512. Let me start by recalling the formula for the sum of the first n terms of a geometric sequence.

The formula is S_n = a_1 * (1 - r^n) / (1 - r), where a_1 is the first term, r is the common ratio, and n is the number of terms. Let me verify if I remember this correctly. Yeah, I think that's right. The sum of a geometric series is calculated using that formula when the common ratio r is not equal to 1.

First, let me identify the first term a_1 and the common ratio r from the given sequence. The sequence is {1/4, 1/8, 1/16, ...}. So the first term a_1 is 1/4. To find the common ratio r, I can divide the second term by the first term. Let me do that: (1/8) divided by (1/4) is equal to (1/8) * (4/1) = 4/8 = 1/2. Let me check with the next terms to make sure. The third term is 1/16, so dividing the third term by the second term: (1/16) / (1/8) = (1/16) * (8/1) = 8/16 = 1/2. Yep, so the common ratio r is 1/2.

So now I have a_1 = 1/4, r = 1/2, and the sum S_n is given as 255/512. I need to solve for n. Let me plug these values into the formula:

S_n = (1/4) * (1 - (1/2)^n) / (1 - 1/2)

Let me simplify the denominator first. 1 - 1/2 is 1/2. So the formula becomes:

S_n = (1/4) * (1 - (1/2)^n) / (1/2)

Dividing by 1/2 is the same as multiplying by 2, so:

S_n = (1/4) * 2 * (1 - (1/2)^n) = (1/2) * (1 - (1/2)^n)

So S_n = (1/2) * [1 - (1/2)^n]. And this is equal to 255/512. Let me write that equation:

(1/2) * [1 - (1/2)^n] = 255/512

To solve for n, I can first multiply both sides by 2 to eliminate the 1/2:

1 - (1/2)^n = (255/512) * 2 = 255/256

Wait, let me check that calculation. 255/512 multiplied by 2 is 255/256? Let me compute 255/512 * 2. 2 divided by 512 is 1/256, so 255 * 2 / 512 = 510/512, which simplifies to 255/256. Yes, that's correct. So:

1 - (1/2)^n = 255/256

Now, let me solve for (1/2)^n. Subtract 255/256 from both sides:

1 - 255/256 = (1/2)^n

Calculating the left side: 1 is 256/256, so 256/256 - 255/256 = 1/256. Therefore:

(1/2)^n = 1/256

Now, I need to solve for n in the equation (1/2)^n = 1/256. Let me recall that (1/2)^n is the same as 2^(-n). Similarly, 1/256 is 2^(-8) because 2^8 is 256. Let me verify that: 2^10 is 1024, so 2^8 is 256. Yes, correct. Therefore:

2^(-n) = 2^(-8)

Since the bases are equal, the exponents must be equal. Therefore:

-n = -8 => n = 8

Wait, so n is 8? Let me check if this makes sense. Let me compute the sum of the first 8 terms and see if it equals 255/512.

First, let me compute S_8 using the formula. S_n = (1/2) * [1 - (1/2)^n]. Plugging in n=8:

S_8 = (1/2) * [1 - (1/2)^8] = (1/2) * [1 - 1/256] = (1/2) * (255/256) = 255/512.

Yes! That's exactly the value given in the problem. So n=8 is the correct answer.

But let me just make sure I didn't make any mistakes in my steps. Let me go through again:

1. Identified a_1 = 1/4, r = 1/2. Correct.

2. Applied the formula S_n = a_1*(1 - r^n)/(1 - r). Correct.

3. Plugged in the values: (1/4)*(1 - (1/2)^n)/(1 - 1/2). Correct.

4. Simplified denominator to 1/2, so multiplied by 2: (1/4)*2*(1 - (1/2)^n) = 1/2*(1 - (1/2)^n). Correct.

5. Set equal to 255/512: 1/2*(1 - (1/2)^n) = 255/512. Correct.

6. Multiply both sides by 2: 1 - (1/2)^n = 255/256. Correct.

7. Subtract 255/256 from 1: 1/256. Correct.

8. So (1/2)^n = 1/256. Which is 2^(-n) = 2^(-8). Therefore, n=8. Correct.

9. Checked by plugging back in: S_8 = 255/512. Correct.

Therefore, the answer is n=8.

But just to be thorough, let me compute the sum manually for n=8 and see if it's indeed 255/512. Let's compute each term:

First term: 1/4 = 1/4

Second term: 1/8

Third term: 1/16

Fourth term: 1/32

Fifth term: 1/64

Sixth term: 1/128

Seventh term: 1/256

Eighth term: 1/512

Now, let's add them up:

Let me convert all to denominators of 512 to add them:

1/4 = 128/512

1/8 = 64/512

1/16 = 32/512

1/32 = 16/512

1/64 = 8/512

1/128 = 4/512

1/256 = 2/512

1/512 = 1/512

Now, adding all numerators:

128 + 64 = 192

192 + 32 = 224

224 + 16 = 240

240 + 8 = 248

248 + 4 = 252

252 + 2 = 254

254 + 1 = 255

So total is 255/512. Perfect! That matches the given sum. Therefore, n=8 is indeed the correct answer.

I think that's solid. I don't see any mistakes in my reasoning. So the answer is 8.

**Final Answer**
The value of $ n $ is \boxed{8}.

To solve the problem, we are given a geometric sequence:

$$
\left\{\frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots \right\}
$$

and told that the sum of the first $ n $ terms is:

$$
\frac{255}{512}
$$

We are to find the value of $ n $.

---

### Step 1: Identify the first term and common ratio

From the sequence:

- First term: $ a_1 = \frac{1}{4} $
- Common ratio: $ r = \frac{1/8}{1/4} = \frac{1}{2} $

---

### Step 2: Use the formula for the sum of the first $ n $ terms of a geometric sequence

The formula is:

$$
S_n = a_1 \cdot \frac{1 - r^n}{1 - r}
$$

Substituting the known values:

$$
S_n = \frac{1}{4} \cdot \frac{1 - \left(\frac{1}{2}\right)^n}{1 - \frac{1}{2}} = \frac{1}{4} \cdot \frac{1 - \left(\frac{1}{2}\right)^n}{\frac{1}{2}} = \frac{1}{4} \cdot 2 \cdot \left(1 - \left(\frac{1}{2}\right)^n\right)
$$

Simplify:

$$
S_n = \frac{1}{2} \left(1 - \left(\frac{1}{2}\right)^n\right)
$$

We are told that this equals $ \frac{255}{512} $, so:

$$
\frac{1}{2} \left(1 - \left(\frac{1}{2}\right)^n\right) = \frac{255}{512}
$$

---

### Step 3: Solve for $ n $

Multiply both sides by 2:

$$
1 - \left(\frac{1}{2}\right)^n = \frac{255}{256}
$$

Subtract $ \frac{255}{256} $ from 1:

$$
\left(\frac{1}{2}\right)^n = 1 - \frac{255}{256} = \frac{1}{256}
$$

Now, express both sides as powers of 2:

$$
\left(\frac{1}{2}\right)^n = 2^{-n}, \quad \frac{1}{256} = 2^{-8}
$$

So:

$$
2^{-n} = 2^{-8} \Rightarrow n = 8
$$

---

### Step 4: Verify the result

Let’s compute the sum of the first 8 terms using the formula:

$$
S_8 = \frac{1}{2} \left(1 - \left(\frac{1}{2}\right)^8\right) = \frac{1}{2} \left(1 - \frac{1}{256}\right) = \frac{1}{2} \cdot \frac{255}{256} = \frac{255}{512}
$$

This matches the given sum, confirming that $ n = 8 $ is correct.

---

### Final Answer

$$
\boxed{8}
$$

Solution:
This is a geometric sequence with first term $\frac{1}{4}$ and common ratio $\frac{1}{2}$. Thus the sum of the first $n$ terms is:

$\frac{255}{512}=\frac{1}{4}\left(\frac{1-\left(\frac{1}{2}\right)^n}{1-\frac{1}{2}}\right)=\frac{2^n-1}{2^{n+1}}$.

We see that $\frac{255}{512}=\frac{2^8-1}{2^9}$, so $n=\boxed{8}$.