Explanation:
Okay, let me try to figure out this problem. So, there are twelve friends who went to dinner, and each ordered one meal. But the portions were so large that there was enough food for 18 people. Now, the question is, if they share, how many meals should they have ordered to have just enough food for the 12 of them?

Hmm, let me break this down. First, they ordered 12 meals, but the food was enough for 18 people. That means each meal they ordered actually provided enough food for more than one person. Wait, how does that work? Maybe each meal is portioned for a certain number of people? Let me think.

If 12 meals were enough for 18 people, then each meal must be providing food for 18 divided by 12 people. Let me write that down. So, the amount of food per meal is 18 people / 12 meals. Let me calculate that. 18 divided by 12 is 1.5. So each meal is equivalent to 1.5 people's worth of food? That seems like a lot, but maybe the portions are really large.

So, if each meal is enough for 1.5 people, then if they want to have just enough food for 12 people, how many meals do they need? Let me think. If one meal is 1.5 people, then the number of meals needed for 12 people would be 12 divided by 1.5. Let me do that calculation. 12 divided by 1.5. Hmm, 1.5 times 8 is 12, right? Because 1.5 times 2 is 3, times 4 is 6, times 8 is 12. So that would be 8 meals. Wait, so they should have ordered 8 meals instead of 12?

But let me check again. If 12 meals are enough for 18 people, then the total amount of food is equivalent to 18 people. So if they want to have just enough food for 12 people, they need to order meals such that the total food equals 12 people. Since 12 meals give 18 people, then the ratio of meals to people is 12 meals : 18 people. So, simplifying that ratio, divide both by 6: 2 meals : 3 people. So, for every 2 meals, you get food for 3 people. Therefore, to get food for 12 people, how many meals?

Set up a proportion. Let me write that as:

2 meals / 3 people = x meals / 12 people

Cross-multiplying, 3x = 2 * 12 => 3x = 24 => x = 8. So that's 8 meals. So that seems to confirm the earlier answer. So they should have ordered 8 meals.

Alternatively, maybe think in terms of how much food each person gets. If 12 meals are enough for 18 people, then each person gets 12 meals / 18 people = 2/3 of a meal per person. Wait, that seems different. Wait, maybe I need to think about it differently.

Wait, if there are 12 meals, and they are enough for 18 people, then each person gets 12 meals divided by 18 people. So that's 12/18 = 2/3 of a meal per person. So if they want to have just enough food for 12 people, each person should get 1 meal. So total meals needed would be 12 people * 1 meal per person = 12 meals? Wait, that contradicts the previous answer. Wait, so which is it?

Wait, maybe I confused something here. Let me think again. If each meal is large enough for multiple people, then the total amount of food is 12 meals. But that food is enough for 18 people. So the total food is equivalent to 18 people's worth. Therefore, if they want to have food for 12 people, they need to have 12 people's worth of food. Since 12 meals = 18 people, then 1 person's worth of food is 12 meals / 18 = 2/3 meals. Therefore, 12 people would need 12 * (2/3) = 8 meals. Ah, there we go. So that's the same answer as before. So that's 8 meals. So the confusion was in the second approach where I thought each person gets 2/3 of a meal, but actually, the total food required for 12 people is 12*(2/3) = 8 meals. So that's correct.

Alternatively, maybe think of it as scaling. If 12 meals are enough for 18 people, then to find out how many meals are needed for 12 people, we can set up a proportion. Let me write that:

12 meals / 18 people = x meals / 12 people

Wait, but that would be if we are keeping the same ratio. Wait, but actually, if you want to find x meals that would be equivalent to 12 people, given that 12 meals is equivalent to 18 people. So, the ratio of meals to people is 12/18 = 2/3. Therefore, to get 12 people, you need x meals such that x / 12 = 2/3. Wait, no, that would be if meals per person is 2/3. Wait, maybe I need to think of it as:

If 12 meals = 18 people, then 1 meal = 18/12 people = 1.5 people. Therefore, to get 12 people, you need 12 / 1.5 = 8 meals. So that's the same answer again.

So all methods lead to 8 meals. Therefore, the answer is 8. But let me check once more with another approach.

Suppose each meal is designed for 1.5 people. So if you have one meal, you can feed 1.5 people. Therefore, to feed 12 people, you need 12 divided by 1.5. Let me compute that. 1.5 goes into 12 how many times? 1.5 * 8 = 12. So yes, 8 meals. So that's consistent.

Alternatively, think of it as fractions. If 12 meals = 18 people, then 1 person = 12/18 meals = 2/3 meals. Therefore, 12 people = 12 * 2/3 = 8 meals. Yep, same answer.

So all methods lead to 8. Therefore, the answer is 8. Therefore, they should have ordered 8 meals.

But let me think again if there's another way someone might approach this problem and get confused. For example, someone might think that since there was enough food for 18 people, but they only had 12, so they had 12/18 = 2/3 of the food. Therefore, to have just enough, they need to order 12 people's worth, which would be 12 divided by (2/3) = 18? Wait, that would be wrong. Wait, why?

Wait, if they ordered 12 meals, which was enough for 18 people, so the total food is 18 people's worth. If they only wanted to have food for 12 people, they would need to order less. So if they had ordered 12 meals, they had 18 people's food. Therefore, to have 12 people's food, they need to order (12/18)*12 meals? Wait, that would be 8 meals. Wait, that's the same answer. Wait, so if the total food is proportional to the number of meals. So if you want to have 12 people's food, which is (12/18) of the original food, then you need to order (12/18)*12 meals? Wait, no, that would be if you wanted to have 12 people's food, but the original 12 meals was 18 people's food. So the amount of food is proportional to the number of meals. Therefore, if you want to have food for 12 people, which is (12/18) of the original food, then you need to order (12/18)*original number of meals. Wait, original number of meals was 12. So (12/18)*12 = 8. So that's again 8 meals. So that's another way to think about it.

Alternatively, if you think of it as scaling factor. The original number of meals was 12, which was enough for 18 people. So if you want to scale down the number of meals to be enough for 12 people, you need to multiply by the ratio of 12/18. So 12 * (12/18) = 8. So that's the same answer.

So all these different approaches lead to 8. Therefore, I think the answer is 8. Therefore, they should have ordered 8 meals.

But let me check once more with actual numbers. Suppose each meal is 1.5 people. So if you have 8 meals, that's 8 * 1.5 = 12 people. Exactly. So that's correct. So if they ordered 8 meals, that would be enough for 12 people. Therefore, the answer is 8.

Alternatively, if they ordered 12 meals, that's 12 * 1.5 = 18 people. Which matches the problem statement. So that's correct. Therefore, the answer is 8.

I think that's solid. All different ways of thinking about it lead to 8. So I think that's the answer.

**Final Answer**
The twelve friends should have ordered \boxed{8} meals.

To determine how many meals the twelve friends should have ordered to have just enough food for themselves, we begin by analyzing the given information:

- **12 meals** were ordered.
- These meals were **enough for 18 people**.

This implies that the **total amount of food** from 12 meals is equivalent to the food needed for 18 people. Therefore, we can calculate how much food each meal provides in terms of "person equivalents."

---

### Step 1: Determine the food per meal

We calculate the amount of food per meal by dividing the total number of people the food can feed by the number of meals:

$$
\text{Food per meal} = \frac{18 \text{ people}}{12 \text{ meals}} = 1.5 \text{ people per meal}
$$

So, **each meal is enough for 1.5 people**.

---

### Step 2: Calculate the number of meals needed for 12 people

Now, we want to find out how many meals are needed to feed **12 people**, given that each meal provides food for 1.5 people.

$$
\text{Number of meals} = \frac{12 \text{ people}}{1.5 \text{ people per meal}} = 8 \text{ meals}
$$

---

### Step 3: Verify the result

Let’s verify this by checking the total food amount:

- 8 meals × 1.5 people per meal = 12 people

This confirms that **8 meals** are sufficient to feed **12 people**.

---

### Final Answer

$$
\boxed{8}
$$

[Assistant]
Explanation:
Okay, let me try to figure out this problem. So, there are twelve friends who went to dinner, and each ordered one meal. But the portions were so large that there was enough food for 18 people. Now, the question is, if they share, how many meals should they have ordered to have just enough food for the 12 of them?

Hmm, let me break this down. First, they ordered 12 meals, but the food was enough for 18 people. That means each meal they ordered actually provided enough food for more than one person. Wait, how does that work? Maybe each meal is portioned for a certain number of people? Let me think.

If 12 meals were enough for 18 people, then each meal must be providing food for 18 divided by 12 people. Let me write that down. So, the amount of food per meal is 18 people / 12 meals. Let me calculate that. 18 divided by 12 is 1.5. So each meal is equivalent to 1.5 people's worth of food? That seems like a lot, but maybe the portions are really large.

So, if each meal is enough for 1.5 people, then if they want to have just enough food for 12 people, how many meals do they need? Let me think. If one meal is 1.5 people, then the number of meals needed for 12 people would be 12 divided by 1.5. Let me do that calculation. 12 divided by 1.5. Hmm, 1.5 times 8 is 12, right? Because 1.5 times 2 is 3, times 4 is 6, times 8 is 12. So that would be 8 meals. Wait, so they should have ordered 8 meals instead of 12?

But let me check again. If 12 meals are enough for 18 people, then the total amount of food is equivalent to 18 people. So if they want to have just enough food for 12 people, they need to order meals such that the total food equals 12 people. Since 12 meals give 18 people, then the ratio of meals to people is 12 meals : 18 people. So, simplifying that ratio, divide both by 6: 2 meals : 3 people. So, for every 2 meals, you get food for 3 people. Therefore, to get food for 12 people, how many meals?

Set up a proportion. Let me write that as:

2 meals / 3 people = x meals / 12 people

Cross-multiplying, 3x = 2 * 12 => 3x = 24 => x = 8. So that's 8 meals. So that seems to confirm the earlier answer. So they should have ordered 8 meals.

Alternatively, maybe think in terms of how much food each person gets. If 12 meals are enough for 18 people, then each person gets 12 meals / 18 people = 2/3 of a meal per person. Wait, that seems different. Wait, maybe I need to think about it differently.

Wait, if there are 12 meals, and they are enough for 18 people, then each person gets 12 meals divided by 18 people. So that's 12/18 = 2/3 of a meal per person. So if they want to have just enough food for 12 people, each person should get 1 meal. So total meals needed would be 12 people * 1 meal per person = 12 meals? Wait, that contradicts the previous answer. Wait, so which is it?

Wait, maybe I confused something here. Let me think again. If each meal is large enough for multiple people, then the total amount of food is 12 meals. But that food is enough for 18 people. So the total food is equivalent to 18 people's worth. Therefore, if they want to have food for 12 people, they need to have 12 people's worth of food. Since 12 meals = 18 people, then 1 person's worth of food is 12 meals / 18 = 2/3 meals. Therefore, 12 people would need 12 * (2/3) = 8 meals. Ah, there we go. So that's the same answer as before. So that's 8 meals. So the confusion was in the second approach where I thought each person gets 2/3 of a meal, but actually, the total food required for 12 people is 12*(2/3) = 8 meals. So that's correct.

Alternatively, maybe think of it as scaling. If 12 meals are enough for 18 people, then to find out how many meals are needed for 12 people, we can set up a proportion. Let me write that:

12 meals / 18 people = x meals / 12 people

Wait, but that would be if we are keeping the same ratio. Wait, but actually, if you want to find x meals that would be equivalent to 12 people, given that 12 meals is equivalent to 18 people. So, the ratio of meals to people is 12/18 = 2/3. Therefore, to get 12 people, you need x meals such that x / 12 = 2/3. Wait, no, that would be if meals per person is 2/3. Wait, maybe I need to think of it as:

If 12 meals = 18 people, then 1 meal = 18/12 people = 1.5 people. Therefore, to get 12 people, you need 12 / 1.5 = 8 meals. So that's the same answer again.

So all methods lead to 8 meals. Therefore, the answer is 8. But let me check once more with another approach.

Suppose each meal is designed for 1.5 people. So if you have one meal, you can feed 1.5 people. Therefore, to feed 12 people, you need 12 divided by 1.5. Let me compute that. 1.5 goes into 12 how many times? 1.5 * 8 = 12. So yes, 8 meals. So that's consistent.

Alternatively, think of it as fractions. If 12 meals = 18 people, then 1 person = 12/18 meals = 2/3 meals. Therefore, 12 people = 12 * 2/3 = 8 meals. Yep, same answer.

So all methods lead to 8. Therefore, the answer is 8. Therefore, they should have ordered 8 meals.

But let me think again if there's another way someone might approach this problem and get confused. For example, someone might think that since there was enough food for 18 people, but they only had 12, so they had 12/18 = 2/3 of the food. Therefore, to have just enough, they need to order 12 people's worth, which would be 12 divided by (2/3) = 18? Wait, that would be wrong. Wait, why?

Wait, if they ordered 12 meals, which was enough for 18 people, so the total food is 18 people's worth. If they only wanted to have food for 12 people, they would need to order less. So if they had ordered 12 meals, they had 18 people's food. Therefore, to have 12 people's food, they need to order (12/18)*12 meals? Wait, that would be 8 meals. Wait, that's the same answer. Wait, so if the total food is proportional to the number of meals. So if you want to have 12 people's food, which is (12/18) of the original food, then you need to order (12/18)*12 meals? Wait, no, that would be if you wanted to have 12 people's food, but the original 12 meals was 18 people's food. So the amount of food is proportional to the number of meals. Therefore, if you want to have food for 12 people, which is (12/18) of the original food, then you need to order (12/18)*original number of meals. Wait, original number of meals was 12. So (12/18)*12 = 8. So that's again 8 meals. So that's another way to think about it.

Alternatively, if you think of it as scaling factor. The original number of meals was 12, which was enough for 18 people. So if you want to scale down the number of meals to be enough for 12 people, you need to multiply by the ratio of 12/18. So 12 * (12/18) = 8. So that's the same answer.

So all these different approaches lead to 8. Therefore, I think the answer is 8. Therefore, they should have ordered 8 meals.

But let me check once more with actual numbers. Suppose each meal is 1.5 people. So if you have 8 meals, that's 8 * 1.5 = 12 people. Exactly. So that's correct. So if they ordered 8 meals, that would be enough for 12 people. Therefore, the answer is 8.

Alternatively, if they ordered 12 meals, that's 12 * 1.5 = 18 people. Which matches the problem statement. So that's correct. Therefore, the answer is 8.

I think that's solid. All different ways of thinking about it lead to 8. So I think that's the answer.

**Final Answer**
The twelve friends should have ordered \boxed{8} meals.

To determine how many meals the twelve friends should have ordered to have just enough food for themselves, we begin by analyzing the given information:

- **12 meals** were ordered.
- These meals were **enough for 18 people**.

This implies that the **total amount of food** from 12 meals is equivalent to the food needed for 18 people. Therefore, we can calculate how much food each meal provides in terms of "person equivalents."

---

### Step 1: Determine the food per meal

We calculate the amount of food per meal by dividing the total number of people the food can feed by the number of meals:

$$
\text{Food per meal} = \frac{18 \text{ people}}{12 \text{ meals}} = 1.5 \text{ people per meal}
$$

So, **each meal is enough for 1.5 people**.

---

### Step 2: Calculate the number of meals needed for 12 people

Now, we want to find out how many meals are needed to feed **12 people**, given that each meal provides food for 1.5 people.

$$
\text{Number of meals} = \frac{12 \text{ people}}{1.5 \text{ people per meal}} = 8 \text{ meals}
$$

---

### Step 3: Verify the result

Let’s verify this by checking the total food amount:

- 8 meals × 1.5 people per meal = 12 people

This confirms that **8 meals** are sufficient to feed **12 people**.

---

### Final Answer

$$
\boxed{8}
$$

Solution: If 12 people order $\frac{18}{12}=1\frac{1}{2}$ times too much food, they should have ordered $\frac{12}{\frac{3}{2}}=\frac{2}{3}\times 12=\boxed{8}$ meals.


Explanation:
Okay, so I need to figure out what kind of curve is defined by the equation $ r^2 \cos 2\theta = 4 $. The options are Line, Circle, Parabola, Ellipse, or Hyperbola. Let me start by recalling some polar equations and their corresponding curves.

First, I remember that in polar coordinates, equations can represent various conic sections, but I need to recall the specific forms. Also, the equation given here has $ r^2 $ and $ \cos 2\theta $, which makes me think it might be related to some kind of conic section or maybe a different curve. Let me try to manipulate the equation to see if I can convert it into Cartesian coordinates, which might be easier to identify.

So, the original equation is $ r^2 \cos 2\theta = 4 $. Let me recall that $ \cos 2\theta $ can be expressed in terms of $ x $ and $ y $. Also, $ r^2 = x^2 + y^2 $. Let me write down those identities:

First, $ \cos 2\theta $ can be written using the double angle formula. There are a few forms:

$ \cos 2\theta = \cos^2 \theta - \sin^2 \theta $, or $ 2\cos^2 \theta - 1 $, or $ 1 - 2\sin^2 \theta $. Alternatively, in terms of $ x $ and $ y $, since $ x = r \cos \theta $ and $ y = r \sin \theta $, so $ \cos \theta = x/r $ and $ \sin \theta = y/r $. Therefore, $ \cos 2\theta = \cos^2 \theta - \sin^2 \theta = (x^2 - y^2)/r^2 $.

So substituting that into the original equation:

$ r^2 \times \frac{x^2 - y^2}{r^2} = 4 $

Simplifying that, the $ r^2 $ cancels out:

$ x^2 - y^2 = 4 $

Wait, so that simplifies to $ x^2 - y^2 = 4 $. That looks like a hyperbola. Because the standard form of a hyperbola is $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $. Comparing that with $ x^2 - y^2 = 4 $, we can divide both sides by 4 to get $ \frac{x^2}{4} - \frac{y^2}{4} = 1 $, which is indeed a hyperbola with $ a^2 = 4 $ and $ b^2 = 4 $.

But let me double-check my steps to make sure I didn't make a mistake. Starting with $ r^2 \cos 2\theta = 4 $. Then, using the identity $ \cos 2\theta = \frac{x^2 - y^2}{r^2} $. Therefore, substituting that into the equation gives $ r^2 \times \frac{x^2 - y^2}{r^2} = x^2 - y^2 = 4 $. Yes, that seems correct.

Alternatively, maybe I can think about other forms. For example, sometimes polar equations can represent conic sections with the focus at the origin. The general form for a conic in polar coordinates is $ r = \frac{ed}{1 + e \cos \theta} $ or similar, where $ e $ is the eccentricity. But in this case, the equation is $ r^2 \cos 2\theta = 4 $, which is not in that standard form. So maybe converting to Cartesian coordinates is the better approach here.

Alternatively, maybe I can think about the equation $ x^2 - y^2 = 4 $. Let me recall that this is a hyperbola centered at the origin, opening along the x-axis. So that would be a hyperbola. Therefore, the answer should be (E) Hyperbola.

But let me check if there's another way to think about this. For example, sometimes equations in polar coordinates can represent other curves. Let me think if there's a possibility that this could be something else. For example, if I consider $ r^2 \cos 2\theta = 4 $, maybe I can write it as $ r^2 = 4 / \cos 2\theta $, which is $ r^2 = 4 \sec 2\theta $. But I don't recall a standard polar equation that looks like this. However, converting to Cartesian coordinates seems more straightforward and gives a hyperbola.

Alternatively, maybe I can think about specific points. For example, when $ \theta = 0 $, then $ \cos 0 = 1 $, so the equation becomes $ r^2 * 1 = 4 $, so $ r = \pm 2 $. So the points (2, 0) and (-2, 0) are on the curve. Similarly, when $ \theta = \pi/4 $, then $ \cos 2\theta = \cos (\pi/2) = 0 $, so the equation becomes $ r^2 * 0 = 4 $, which is 0 = 4, which is impossible. Therefore, there are no points at $ \theta = \pi/4 $. Similarly, when $ \theta = \pi/2 $, $ \cos 2\theta = \cos \pi = -1 $, so the equation becomes $ r^2 * (-1) = 4 $, so $ r^2 = -4 $, which is impossible in real numbers. Therefore, there are no points at $ \theta = \pi/2 $.

So, the curve doesn't exist at angles where $ \cos 2\theta $ is negative. Therefore, the curve exists only where $ \cos 2\theta $ is positive, which is when $ 2\theta $ is in the first or fourth quadrants, i.e., $ \theta $ in the first or fourth quadrants, between $ -\pi/4 $ to $ \pi/4 $, and then between $ 3\pi/4 $ to $ 5\pi/4 $, etc. Wait, actually, $ \cos 2\theta $ is positive when $ 2\theta $ is in the first or fourth quadrants, so $ 2\theta \in (-\pi/2 + 2\pi k, \pi/2 + 2\pi k) $, so $ \theta \in (-\pi/4 + \pi k, \pi/4 + \pi k) $. Therefore, the curve exists in those angular regions.

But how does this relate to the Cartesian equation? Well, since we already converted it to $ x^2 - y^2 = 4 $, which is a hyperbola, which has two branches. So, in polar coordinates, the equation $ r^2 \cos 2\theta = 4 $ would represent both branches of the hyperbola.

Alternatively, maybe I can think about the general form of conic sections in polar coordinates. But since the equation isn't in the standard conic form, maybe converting to Cartesian is better.

Another way to check is to recall that $ x^2 - y^2 = 4 $ is a hyperbola. Let me confirm that. For example, if I consider the standard hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $, then yes, this is a hyperbola with a = 2 and b = 2. Therefore, it's a rectangular hyperbola because a = b.

Therefore, the answer should be (E) Hyperbola.

But let me check if there's any chance that this could be something else. For example, if I had $ r \cos 2\theta = 4 $, that would be different. But since it's $ r^2 \cos 2\theta = 4 $, which converts to $ x^2 - y^2 = 4 $, which is definitely a hyperbola.

Alternatively, maybe I can think about the polar equation $ r^2 \cos 2\theta = 4 $. Let me recall that $ \cos 2\theta $ can also be expressed as $ \frac{x^2 - y^2}{r^2} $, so substituting that gives $ r^2 * \frac{x^2 - y^2}{r^2} = x^2 - y^2 = 4 $. So that's the same as before.

Alternatively, maybe I can think about parametric equations. Suppose I let $ r^2 = 4 / \cos 2\theta $, so $ r = \sqrt{4 / \cos 2\theta} $. But this seems more complicated.

Alternatively, maybe I can think about the equation in terms of rectangular coordinates. Since $ x^2 - y^2 = 4 $, which is a hyperbola. Therefore, the answer is E.

Wait, but let me check if there's a possibility that the original equation could represent something else. For example, if I consider $ r^2 \cos 2\theta = 4 $, maybe there's a different interpretation. For example, sometimes equations can be manipulated in different ways. Let me think.

Alternatively, maybe using the identity $ \cos 2\theta = \frac{r^2 - 2x^2}{r^2} $? Wait, no. Wait, $ \cos 2\theta = \cos^2 \theta - \sin^2 \theta = \frac{x^2}{r^2} - \frac{y^2}{r^2} = \frac{x^2 - y^2}{r^2} $. So that's correct. Therefore, substituting into the original equation gives $ r^2 * \frac{x^2 - y^2}{r^2} = x^2 - y^2 = 4 $.

Therefore, I think my conversion is correct. Therefore, the equation is a hyperbola.

Alternatively, maybe I can think about the graph. For example, if I consider $ x^2 - y^2 = 4 $, then it's a hyperbola opening left and right. So, in polar coordinates, this would correspond to points where $ r^2 \cos 2\theta = 4 $.

Alternatively, maybe I can think about specific points. For example, when $ \theta = 0 $, as before, $ r^2 = 4 $, so $ r = 2 $, so the point (2, 0). Similarly, when $ \theta = \pi $, $ \cos 2\theta = \cos 2\pi = 1 $, so again $ r^2 = 4 $, so $ r = 2 $, so the point (-2, 0). When $ \theta = \pi/3 $, then $ 2\theta = 2\pi/3 $, so $ \cos 2\pi/3 = -1/2 $. Therefore, the equation becomes $ r^2 * (-1/2) = 4 $, so $ r^2 = -8 $, which is not possible. Therefore, there are no points at $ \theta = \pi/3 $.

But if I take $ \theta = \pi/6 $, then $ 2\theta = \pi/3 $, so $ \cos \pi/3 = 1/2 $, so $ r^2 * 1/2 = 4 $, so $ r^2 = 8 $, so $ r = 2\sqrt{2} $. Therefore, the point in polar coordinates is $ (2\sqrt{2}, \pi/6) $. Converting that to Cartesian coordinates: $ x = r \cos \theta = 2\sqrt{2} \cos \pi/6 = 2\sqrt{2} * (\sqrt{3}/2) = \sqrt{6} $, and $ y = 2\sqrt{2} \sin \pi/6 = 2\sqrt{2} * 1/2 = \sqrt{2} $. Then, $ x^2 - y^2 = 6 - 2 = 4 $, which matches the equation.

So that point is on the hyperbola. Similarly, if I take another angle, say $ \theta = \pi/4 $, but we saw that $ \cos 2\theta = 0 $, so no solution. If I take $ \theta = \pi/12 $, then $ 2\theta = \pi/6 $, so $ \cos \pi/6 = \sqrt{3}/2 $, so $ r^2 * \sqrt{3}/2 = 4 $, so $ r^2 = 8/\sqrt{3} $, so $ r = \sqrt{8/\sqrt{3}} $. Converting that to Cartesian coordinates would give a point that satisfies $ x^2 - y^2 = 4 $.

Therefore, all these points lie on the hyperbola $ x^2 - y^2 = 4 $, which confirms that the original polar equation represents a hyperbola.

Therefore, the answer should be (E) Hyperbola.

**Final Answer**
\boxed{E}

To determine the type of curve defined by the polar equation

$$
r^2 \cos 2\theta = 4,
$$

we begin by converting it into Cartesian coordinates, which is often more intuitive for identifying conic sections.

---

### Step 1: Use Polar to Cartesian Identities

Recall the following identities:

- $ r^2 = x^2 + y^2 $
- $ \cos 2\theta = \frac{x^2 - y^2}{r^2} $

Substitute these into the given equation:

$$
r^2 \cdot \cos 2\theta = r^2 \cdot \frac{x^2 - y^2}{r^2} = x^2 - y^2
$$

So the original equation becomes:

$$
x^2 - y^2 = 4
$$

---

### Step 2: Identify the Curve

The equation $ x^2 - y^2 = 4 $ is a standard form of a **hyperbola**. To see this more clearly, divide both sides by 4:

$$
\frac{x^2}{4} - \frac{y^2}{4} = 1
$$

This is the standard form of a hyperbola centered at the origin, opening along the x-axis, with:

- $ a^2 = 4 $, so $ a = 2 $
- $ b^2 = 4 $, so $ b = 2 $

This is a **rectangular hyperbola**, since $ a = b $.

---

### Step 3: Confirm with Polar Interpretation

We can also interpret the original polar equation $ r^2 \cos 2\theta = 4 $ geometrically. The term $ \cos 2\theta $ introduces symmetry about the x-axis and y-axis, and the presence of $ r^2 $ suggests that the curve is symmetric in both directions from the origin. This is consistent with the behavior of a hyperbola, which has two branches symmetric about the origin.

Additionally, we can test specific values of $ \theta $ to see where the curve exists:

- When $ \theta = 0 $, $ \cos 2\theta = 1 $, so $ r^2 = 4 $, giving $ r = \pm 2 $, which corresponds to the points $ (2, 0) $ and $ (-2, 0) $.
- When $ \theta = \pi/4 $, $ \cos 2\theta = \cos(\pi/2) = 0 $, so the equation becomes $ 0 = 4 $, which is not possible — hence, no points exist at this angle.
- When $ \theta = \pi/6 $, $ \cos 2\theta = \cos(\pi/3) = 1/2 $, so $ r^2 = 8 $, and the corresponding point lies on the hyperbola.

These observations confirm that the curve is a hyperbola.

---

### Final Conclusion

The equation $ r^2 \cos 2\theta = 4 $, when converted to Cartesian coordinates, becomes $ x^2 - y^2 = 4 $, which is a **hyperbola**.

$$
\boxed{E}
$$

[Assistant]
Explanation:
Okay, so I need to figure out what kind of curve is defined by the equation $ r^2 \cos 2\theta = 4 $. The options are Line, Circle, Parabola, Ellipse, or Hyperbola. Let me start by recalling some polar equations and their corresponding curves.

First, I remember that in polar coordinates, equations can represent various conic sections, but I need to recall the specific forms. Also, the equation given here has $ r^2 $ and $ \cos 2\theta $, which makes me think it might be related to some kind of conic section or maybe a different curve. Let me try to manipulate the equation to see if I can convert it into Cartesian coordinates, which might be easier to identify.

So, the original equation is $ r^2 \cos 2\theta = 4 $. Let me recall that $ \cos 2\theta $ can be expressed in terms of $ x $ and $ y $. Also, $ r^2 = x^2 + y^2 $. Let me write down those identities:

First, $ \cos 2\theta $ can be written using the double angle formula. There are a few forms:

$ \cos 2\theta = \cos^2 \theta - \sin^2 \theta $, or $ 2\cos^2 \theta - 1 $, or $ 1 - 2\sin^2 \theta $. Alternatively, in terms of $ x $ and $ y $, since $ x = r \cos \theta $ and $ y = r \sin \theta $, so $ \cos \theta = x/r $ and $ \sin \theta = y/r $. Therefore, $ \cos 2\theta = \cos^2 \theta - \sin^2 \theta = (x^2 - y^2)/r^2 $.

So substituting that into the original equation:

$ r^2 \times \frac{x^2 - y^2}{r^2} = 4 $

Simplifying that, the $ r^2 $ cancels out:

$ x^2 - y^2 = 4 $

Wait, so that simplifies to $ x^2 - y^2 = 4 $. That looks like a hyperbola. Because the standard form of a hyperbola is $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $. Comparing that with $ x^2 - y^2 = 4 $, we can divide both sides by 4 to get $ \frac{x^2}{4} - \frac{y^2}{4} = 1 $, which is indeed a hyperbola with $ a^2 = 4 $ and $ b^2 = 4 $.

But let me double-check my steps to make sure I didn't make a mistake. Starting with $ r^2 \cos 2\theta = 4 $. Then, using the identity $ \cos 2\theta = \frac{x^2 - y^2}{r^2} $. Therefore, substituting that into the equation gives $ r^2 \times \frac{x^2 - y^2}{r^2} = x^2 - y^2 = 4 $. Yes, that seems correct.

Alternatively, maybe I can think about other forms. For example, sometimes polar equations can represent conic sections with the focus at the origin. The general form for a conic in polar coordinates is $ r = \frac{ed}{1 + e \cos \theta} $ or similar, where $ e $ is the eccentricity. But in this case, the equation is $ r^2 \cos 2\theta = 4 $, which is not in that standard form. So maybe converting to Cartesian coordinates is the better approach here.

Alternatively, maybe I can think about the equation $ x^2 - y^2 = 4 $. Let me recall that this is a hyperbola centered at the origin, opening along the x-axis. So that would be a hyperbola. Therefore, the answer should be (E) Hyperbola.

But let me check if there's another way to think about this. For example, sometimes equations in polar coordinates can represent other curves. Let me think if there's a possibility that this could be something else. For example, if I consider $ r^2 \cos 2\theta = 4 $, maybe I can write it as $ r^2 = 4 / \cos 2\theta $, which is $ r^2 = 4 \sec 2\theta $. But I don't recall a standard polar equation that looks like this. However, converting to Cartesian coordinates seems more straightforward and gives a hyperbola.

Alternatively, maybe I can think about specific points. For example, when $ \theta = 0 $, then $ \cos 0 = 1 $, so the equation becomes $ r^2 * 1 = 4 $, so $ r = \pm 2 $. So the points (2, 0) and (-2, 0) are on the curve. Similarly, when $ \theta = \pi/4 $, then $ \cos 2\theta = \cos (\pi/2) = 0 $, so the equation becomes $ r^2 * 0 = 4 $, which is 0 = 4, which is impossible. Therefore, there are no points at $ \theta = \pi/4 $. Similarly, when $ \theta = \pi/2 $, $ \cos 2\theta = \cos \pi = -1 $, so the equation becomes $ r^2 * (-1) = 4 $, so $ r^2 = -4 $, which is impossible in real numbers. Therefore, there are no points at $ \theta = \pi/2 $.

So, the curve doesn't exist at angles where $ \cos 2\theta $ is negative. Therefore, the curve exists only where $ \cos 2\theta $ is positive, which is when $ 2\theta $ is in the first or fourth quadrants, i.e., $ \theta $ in the first or fourth quadrants, between $ -\pi/4 $ to $ \pi/4 $, and then between $ 3\pi/4 $ to $ 5\pi/4 $, etc. Wait, actually, $ \cos 2\theta $ is positive when $ 2\theta $ is in the first or fourth quadrants, so $ 2\theta \in (-\pi/2 + 2\pi k, \pi/2 + 2\pi k) $, so $ \theta \in (-\pi/4 + \pi k, \pi/4 + \pi k) $. Therefore, the curve exists in those angular regions.

But how does this relate to the Cartesian equation? Well, since we already converted it to $ x^2 - y^2 = 4 $, which is a hyperbola, which has two branches. So, in polar coordinates, the equation $ r^2 \cos 2\theta = 4 $ would represent both branches of the hyperbola.

Alternatively, maybe I can think about the general form of conic sections in polar coordinates. But since the equation isn't in the standard conic form, maybe converting to Cartesian is better.

Another way to check is to recall that $ x^2 - y^2 = 4 $ is a hyperbola. Let me confirm that. For example, if I consider the standard hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $, then yes, this is a hyperbola with a = 2 and b = 2. Therefore, it's a rectangular hyperbola because a = b.

Therefore, the answer should be (E) Hyperbola.

But let me check if there's any chance that this could be something else. For example, if I had $ r \cos 2\theta = 4 $, that would be different. But since it's $ r^2 \cos 2\theta = 4 $, which converts to $ x^2 - y^2 = 4 $, which is definitely a hyperbola.

Alternatively, maybe I can think about the polar equation $ r^2 \cos 2\theta = 4 $. Let me recall that $ \cos 2\theta $ can also be expressed as $ \frac{x^2 - y^2}{r^2} $, so substituting that gives $ r^2 * \frac{x^2 - y^2}{r^2} = x^2 - y^2 = 4 $. So that's the same as before.

Alternatively, maybe I can think about parametric equations. Suppose I let $ r^2 = 4 / \cos 2\theta $, so $ r = \sqrt{4 / \cos 2\theta} $. But this seems more complicated.

Alternatively, maybe I can think about the equation in terms of rectangular coordinates. Since $ x^2 - y^2 = 4 $, which is a hyperbola. Therefore, the answer is E.

Wait, but let me check if there's a possibility that the original equation could represent something else. For example, if I consider $ r^2 \cos 2\theta = 4 $, maybe there's a different interpretation. For example, sometimes equations can be manipulated in different ways. Let me think.

Alternatively, maybe using the identity $ \cos 2\theta = \frac{r^2 - 2x^2}{r^2} $? Wait, no. Wait, $ \cos 2\theta = \cos^2 \theta - \sin^2 \theta = \frac{x^2}{r^2} - \frac{y^2}{r^2} = \frac{x^2 - y^2}{r^2} $. So that's correct. Therefore, substituting into the original equation gives $ r^2 * \frac{x^2 - y^2}{r^2} = x^2 - y^2 = 4 $.

Therefore, I think my conversion is correct. Therefore, the equation is a hyperbola.

Alternatively, maybe I can think about the graph. For example, if I consider $ x^2 - y^2 = 4 $, then it's a hyperbola opening left and right. So, in polar coordinates, this would correspond to points where $ r^2 \cos 2\theta = 4 $.

Alternatively, maybe I can think about specific points. For example, when $ \theta = 0 $, as before, $ r^2 = 4 $, so $ r = 2 $, so the point (2, 0). Similarly, when $ \theta = \pi $, $ \cos 2\theta = \cos 2\pi = 1 $, so again $ r^2 = 4 $, so $ r = 2 $, so the point (-2, 0). When $ \theta = \pi/3 $, then $ 2\theta = 2\pi/3 $, so $ \cos 2\pi/3 = -1/2 $. Therefore, the equation becomes $ r^2 * (-1/2) = 4 $, so $ r^2 = -8 $, which is not possible. Therefore, there are no points at $ \theta = \pi/3 $.

But if I take $ \theta = \pi/6 $, then $ 2\theta = \pi/3 $, so $ \cos \pi/3 = 1/2 $, so $ r^2 * 1/2 = 4 $, so $ r^2 = 8 $, so $ r = 2\sqrt{2} $. Therefore, the point in polar coordinates is $ (2\sqrt{2}, \pi/6) $. Converting that to Cartesian coordinates: $ x = r \cos \theta = 2\sqrt{2} \cos \pi/6 = 2\sqrt{2} * (\sqrt{3}/2) = \sqrt{6} $, and $ y = 2\sqrt{2} \sin \pi/6 = 2\sqrt{2} * 1/2 = \sqrt{2} $. Then, $ x^2 - y^2 = 6 - 2 = 4 $, which matches the equation.

So that point is on the hyperbola. Similarly, if I take another angle, say $ \theta = \pi/4 $, but we saw that $ \cos 2\theta = 0 $, so no solution. If I take $ \theta = \pi/12 $, then $ 2\theta = \pi/6 $, so $ \cos \pi/6 = \sqrt{3}/2 $, so $ r^2 * \sqrt{3}/2 = 4 $, so $ r^2 = 8/\sqrt{3} $, so $ r = \sqrt{8/\sqrt{3}} $. Converting that to Cartesian coordinates would give a point that satisfies $ x^2 - y^2 = 4 $.

Therefore, all these points lie on the hyperbola $ x^2 - y^2 = 4 $, which confirms that the original polar equation represents a hyperbola.

Therefore, the answer should be (E) Hyperbola.

**Final Answer**
\boxed{E}

To determine the type of curve defined by the polar equation

$$
r^2 \cos 2\theta = 4,
$$

we begin by converting it into Cartesian coordinates, which is often more intuitive for identifying conic sections.

---

### Step 1: Use Polar to Cartesian Identities

Recall the following identities:

- $ r^2 = x^2 + y^2 $
- $ \cos 2\theta = \frac{x^2 - y^2}{r^2} $

Substitute these into the given equation:

$$
r^2 \cdot \cos 2\theta = r^2 \cdot \frac{x^2 - y^2}{r^2} = x^2 - y^2
$$

So the original equation becomes:

$$
x^2 - y^2 = 4
$$

---

### Step 2: Identify the Curve

The equation $ x^2 - y^2 = 4 $ is a standard form of a **hyperbola**. To see this more clearly, divide both sides by 4:

$$
\frac{x^2}{4} - \frac{y^2}{4} = 1
$$

This is the standard form of a hyperbola centered at the origin, opening along the x-axis, with:

- $ a^2 = 4 $, so $ a = 2 $
- $ b^2 = 4 $, so $ b = 2 $

This is a **rectangular hyperbola**, since $ a = b $.

---

### Step 3: Confirm with Polar Interpretation

We can also interpret the original polar equation $ r^2 \cos 2\theta = 4 $ geometrically. The term $ \cos 2\theta $ introduces symmetry about the x-axis and y-axis, and the presence of $ r^2 $ suggests that the curve is symmetric in both directions from the origin. This is consistent with the behavior of a hyperbola, which has two branches symmetric about the origin.

Additionally, we can test specific values of $ \theta $ to see where the curve exists:

- When $ \theta = 0 $, $ \cos 2\theta = 1 $, so $ r^2 = 4 $, giving $ r = \pm 2 $, which corresponds to the points $ (2, 0) $ and $ (-2, 0) $.
- When $ \theta = \pi/4 $, $ \cos 2\theta = \cos(\pi/2) = 0 $, so the equation becomes $ 0 = 4 $, which is not possible — hence, no points exist at this angle.
- When $ \theta = \pi/6 $, $ \cos 2\theta = \cos(\pi/3) = 1/2 $, so $ r^2 = 8 $, and the corresponding point lies on the hyperbola.

These observations confirm that the curve is a hyperbola.

---

### Final Conclusion

The equation $ r^2 \cos 2\theta = 4 $, when converted to Cartesian coordinates, becomes $ x^2 - y^2 = 4 $, which is a **hyperbola**.

$$
\boxed{E}
$$

Solution:
From $r^2 \cos 2 \theta = 4,$
\[r^2 (\cos^2 \theta - \sin^2 \theta) = r^2 \cos^2 \theta - r^2 \sin^2 \theta = 4.\]Then $x^2 - y^2 = 4,$ or
\[\frac{x^2}{4} - \frac{y^2}{4} = 1.\]Thus, the graph represents a hyperbola.  The answer is $\boxed{\text{(E)}}.$

[asy]
unitsize(0.5 cm);

pair moo (real t) {
  real r = sqrt(4/Cos(2*t));
  return (r*Cos(t), r*Sin(t));
}

path foo = moo(-44);
real t;

for (t = -44; t <= 44; t = t + 0.1) {
  foo = foo--moo(t);
}

draw(foo,red);
draw(reflect((0,0),(0,1))*(foo),red);

draw((-4,0)--(4,0));
draw((0,-4)--(0,4));

limits((-4,-4),(4,4),Crop);

label("$r^2 \cos 2 \theta = 4$", (6.5,1.5), red);
[/asy]


Explanation:
Okay, so I need to find the 2003rd term of the sequence of odd numbers: 1, 3, 5, 7, and so on. Let me think about how sequences like this work.

First, I remember that the sequence of odd numbers is an arithmetic sequence. That means each term increases by a constant difference. In this case, the difference between each term is 2. Let me confirm that: 3 - 1 = 2, 5 - 3 = 2, 7 - 5 = 2... yeah, that's right. So the common difference (d) is 2.

Now, arithmetic sequences have a formula for the nth term. I think the formula is something like a_n = a_1 + (n - 1)d, where a_n is the nth term, a_1 is the first term, d is the common difference, and n is the term number. Let me check if I remember that correctly.

Let me test it with the first few terms. For n = 1, a_1 = 1 + (1 - 1)*2 = 1 + 0 = 1. That's correct. For n = 2, a_2 = 1 + (2 - 1)*2 = 1 + 2 = 3. Yep, that works. For n = 3, 1 + (3 - 1)*2 = 1 + 4 = 5. Perfect. So the formula seems right.

So applying this formula to find the 2003rd term. Let me write that down:

a_2003 = a_1 + (2003 - 1)*d

We know a_1 is 1, d is 2. So plugging those in:

a_2003 = 1 + (2003 - 1)*2

Let me compute 2003 - 1 first. That's 2002. Then multiply by 2: 2002 * 2. Hmm, 2000*2 is 4000, and 2*2 is 4, so total is 4004. Then add the 1 from the original formula: 1 + 4004 = 4005.

Wait, so is the 2003rd term 4005? Let me check with a smaller term to make sure I didn't mess up the formula. Let's take n = 4. According to the formula, a_4 = 1 + (4 - 1)*2 = 1 + 6 = 7. Which matches the sequence: 1, 3, 5, 7. So that works.

Another check: n = 5. a_5 = 1 + 4*2 = 1 + 8 = 9. Which is correct. So the formula seems to hold.

Alternatively, I remember that the nth odd number is given by 2n - 1. Wait, let me think. If n = 1, 2(1) - 1 = 1. n = 2, 2(2) - 1 = 3. n = 3, 5. Yep, that works. So that formula is another way to think about it. So if that's the case, then the nth term is 2n - 1. Therefore, the 2003rd term would be 2*2003 - 1. Let me compute that.

2*2003 is 4006, minus 1 is 4005. So same answer. So that confirms it.

But why is that formula 2n - 1? Let me think. Because the sequence of odd numbers is 1, 3, 5,... which is the same as starting at 1 and adding 2 each time. So using the arithmetic sequence formula, a_n = a_1 + (n - 1)d. Which is 1 + (n - 1)*2 = 1 + 2n - 2 = 2n - 1. Yep, that's how you get there.

So both methods give me 4005. Therefore, I think that's the answer.

But just to be thorough, let me think if there's another way to approach this. Maybe by considering that each odd number can be represented as 2k - 1 where k is a positive integer. So if I want the 2003rd term, that would correspond to k = 2003. Therefore, the term is 2*2003 - 1 = 4005.

Alternatively, if I think about the relationship between the term number and the value. The first term is 1, which is 2*1 - 1. The second term is 3, which is 2*2 - 1. So yeah, the pattern is clear.

Another way: think about how many numbers are there between 1 and the term we want. But maybe that's complicating.

Alternatively, think about the sum of the first n odd numbers. Wait, but that's a different problem. The sum of the first n odd numbers is n². But that's not directly helpful here. But maybe if I think about it, if the sum of the first n odd numbers is n², then the nth odd number is the difference between the sum of the first n and the sum of the first n-1. So, sum of first n is n², sum of first n-1 is (n - 1)². Therefore, the nth term is n² - (n - 1)². Let me compute that.

n² - (n - 1)² = [n - (n - 1)][n + (n - 1)] = (1)(2n - 1) = 2n - 1. So that's another way to derive the formula. Therefore, the nth term is 2n - 1.

Therefore, again, the 2003rd term is 2*2003 - 1 = 4005.

So all methods lead to the same answer. Therefore, I can be confident that the 2003rd term is 4005.

But just to make sure, let me try to compute a few terms with this formula. For n = 1, 2(1) - 1 = 1. Correct. For n = 2, 3. Correct. For n = 3, 5. Correct. For n = 4, 7. Correct. For n = 5, 9. Correct. So the formula works.

Alternatively, if I think about the sequence of odd numbers, they are all numbers that are not divisible by 2. So the nth odd number is the number that is 2 less than the next even number. Wait, maybe not helpful.

Alternatively, think about the position in the sequence. The first term is 1, which is 2*1 - 1. The second term is 3, which is 2*2 - 1. So yeah, the pattern is clear.

Alternatively, think about the fact that every odd number can be paired with an even number. But maybe that's overcomplicating.

Alternatively, think about the sequence as starting at 1 and adding 2 each time. So the nth term is 1 + 2*(n - 1). Which is the same as 2n - 1.

So again, same formula.

Therefore, regardless of the approach, the answer is 4005.

But just to make sure, let me compute 2*2003. 2000*2 is 4000, 3*2 is 6, so total is 4006. Then subtract 1, so 4005. Yep.

Alternatively, if I think about the 2003rd term, since each term is 2 more than the previous, starting at 1. So term 1: 1, term 2: 1 + 2, term 3: 1 + 2*2, term 4: 1 + 2*3,... term n: 1 + 2*(n - 1). Which again is 2n - 1.

So yeah, all roads lead to 4005.

Therefore, I think the answer is 4005.

**Final Answer**
The 2003rd term of the sequence is \boxed{4005}.

To find the **2003rd term** of the sequence of odd numbers:
$$
1, 3, 5, 7, \dots
$$

we recognize that this is an **arithmetic sequence** with:

- First term $ a_1 = 1 $
- Common difference $ d = 2 $

---

### Step 1: Use the formula for the nth term of an arithmetic sequence

The general formula for the $ n $-th term of an arithmetic sequence is:

$$
a_n = a_1 + (n - 1)d
$$

Substituting the known values:

$$
a_{2003} = 1 + (2003 - 1) \cdot 2
$$

$$
a_{2003} = 1 + 2002 \cdot 2
$$

$$
a_{2003} = 1 + 4004 = 4005
$$

---

### Step 2: Confirm with an alternative formula

Another way to express the $ n $-th odd number is:

$$
a_n = 2n - 1
$$

Using this formula:

$$
a_{2003} = 2 \cdot 2003 - 1 = 4006 - 1 = 4005
$$

---

### Final Answer

$$
\boxed{4005}
$$

[Assistant]
Explanation:
Okay, so I need to find the 2003rd term of the sequence of odd numbers: 1, 3, 5, 7, and so on. Let me think about how sequences like this work.

First, I remember that the sequence of odd numbers is an arithmetic sequence. That means each term increases by a constant difference. In this case, the difference between each term is 2. Let me confirm that: 3 - 1 = 2, 5 - 3 = 2, 7 - 5 = 2... yeah, that's right. So the common difference (d) is 2.

Now, arithmetic sequences have a formula for the nth term. I think the formula is something like a_n = a_1 + (n - 1)d, where a_n is the nth term, a_1 is the first term, d is the common difference, and n is the term number. Let me check if I remember that correctly.

Let me test it with the first few terms. For n = 1, a_1 = 1 + (1 - 1)*2 = 1 + 0 = 1. That's correct. For n = 2, a_2 = 1 + (2 - 1)*2 = 1 + 2 = 3. Yep, that works. For n = 3, 1 + (3 - 1)*2 = 1 + 4 = 5. Perfect. So the formula seems right.

So applying this formula to find the 2003rd term. Let me write that down:

a_2003 = a_1 + (2003 - 1)*d

We know a_1 is 1, d is 2. So plugging those in:

a_2003 = 1 + (2003 - 1)*2

Let me compute 2003 - 1 first. That's 2002. Then multiply by 2: 2002 * 2. Hmm, 2000*2 is 4000, and 2*2 is 4, so total is 4004. Then add the 1 from the original formula: 1 + 4004 = 4005.

Wait, so is the 2003rd term 4005? Let me check with a smaller term to make sure I didn't mess up the formula. Let's take n = 4. According to the formula, a_4 = 1 + (4 - 1)*2 = 1 + 6 = 7. Which matches the sequence: 1, 3, 5, 7. So that works.

Another check: n = 5. a_5 = 1 + 4*2 = 1 + 8 = 9. Which is correct. So the formula seems to hold.

Alternatively, I remember that the nth odd number is given by 2n - 1. Wait, let me think. If n = 1, 2(1) - 1 = 1. n = 2, 2(2) - 1 = 3. n = 3, 5. Yep, that works. So that formula is another way to think about it. So if that's the case, then the nth term is 2n - 1. Therefore, the 2003rd term would be 2*2003 - 1. Let me compute that.

2*2003 is 4006, minus 1 is 4005. So same answer. So that confirms it.

But why is that formula 2n - 1? Let me think. Because the sequence of odd numbers is 1, 3, 5,... which is the same as starting at 1 and adding 2 each time. So using the arithmetic sequence formula, a_n = a_1 + (n - 1)d. Which is 1 + (n - 1)*2 = 1 + 2n - 2 = 2n - 1. Yep, that's how you get there.

So both methods give me 4005. Therefore, I think that's the answer.

But just to be thorough, let me think if there's another way to approach this. Maybe by considering that each odd number can be represented as 2k - 1 where k is a positive integer. So if I want the 2003rd term, that would correspond to k = 2003. Therefore, the term is 2*2003 - 1 = 4005.

Alternatively, if I think about the relationship between the term number and the value. The first term is 1, which is 2*1 - 1. The second term is 3, which is 2*2 - 1. So yeah, the pattern is clear.

Another way: think about how many numbers are there between 1 and the term we want. But maybe that's complicating.

Alternatively, think about the sum of the first n odd numbers. Wait, but that's a different problem. The sum of the first n odd numbers is n². But that's not directly helpful here. But maybe if I think about it, if the sum of the first n odd numbers is n², then the nth odd number is the difference between the sum of the first n and the sum of the first n-1. So, sum of first n is n², sum of first n-1 is (n - 1)². Therefore, the nth term is n² - (n - 1)². Let me compute that.

n² - (n - 1)² = [n - (n - 1)][n + (n - 1)] = (1)(2n - 1) = 2n - 1. So that's another way to derive the formula. Therefore, the nth term is 2n - 1.

Therefore, again, the 2003rd term is 2*2003 - 1 = 4005.

So all methods lead to the same answer. Therefore, I can be confident that the 2003rd term is 4005.

But just to make sure, let me try to compute a few terms with this formula. For n = 1, 2(1) - 1 = 1. Correct. For n = 2, 3. Correct. For n = 3, 5. Correct. For n = 4, 7. Correct. For n = 5, 9. Correct. So the formula works.

Alternatively, if I think about the sequence of odd numbers, they are all numbers that are not divisible by 2. So the nth odd number is the number that is 2 less than the next even number. Wait, maybe not helpful.

Alternatively, think about the position in the sequence. The first term is 1, which is 2*1 - 1. The second term is 3, which is 2*2 - 1. So yeah, the pattern is clear.

Alternatively, think about the fact that every odd number can be paired with an even number. But maybe that's overcomplicating.

Alternatively, think about the sequence as starting at 1 and adding 2 each time. So the nth term is 1 + 2*(n - 1). Which is the same as 2n - 1.

So again, same formula.

Therefore, regardless of the approach, the answer is 4005.

But just to make sure, let me compute 2*2003. 2000*2 is 4000, 3*2 is 6, so total is 4006. Then subtract 1, so 4005. Yep.

Alternatively, if I think about the 2003rd term, since each term is 2 more than the previous, starting at 1. So term 1: 1, term 2: 1 + 2, term 3: 1 + 2*2, term 4: 1 + 2*3,... term n: 1 + 2*(n - 1). Which again is 2n - 1.

So yeah, all roads lead to 4005.

Therefore, I think the answer is 4005.

**Final Answer**
The 2003rd term of the sequence is \boxed{4005}.

To find the **2003rd term** of the sequence of odd numbers:
$$
1, 3, 5, 7, \dots
$$

we recognize that this is an **arithmetic sequence** with:

- First term $ a_1 = 1 $
- Common difference $ d = 2 $

---

### Step 1: Use the formula for the nth term of an arithmetic sequence

The general formula for the $ n $-th term of an arithmetic sequence is:

$$
a_n = a_1 + (n - 1)d
$$

Substituting the known values:

$$
a_{2003} = 1 + (2003 - 1) \cdot 2
$$

$$
a_{2003} = 1 + 2002 \cdot 2
$$

$$
a_{2003} = 1 + 4004 = 4005
$$

---

### Step 2: Confirm with an alternative formula

Another way to express the $ n $-th odd number is:

$$
a_n = 2n - 1
$$

Using this formula:

$$
a_{2003} = 2 \cdot 2003 - 1 = 4006 - 1 = 4005
$$

---

### Final Answer

$$
\boxed{4005}
$$

Solution: The sequence of odd numbers 1, 3, 5, 7, and so on, is an arithmetic sequence, with common difference 2.  Therefore, the $2003^{\text{rd}}$ term is $1+2002\cdot2=\boxed{4005}$.


Explanation:
Okay, so I need to find the value of g(f(5) - 1). Let me start by recalling what the functions f(x) and g(x) are. The problem says f(x) is 2x minus 3, and g(x) is x plus 1. Alright, so first things first, I need to figure out what f(5) is.

Let me write that down. f(x) = 2x - 3. So if I plug in 5 for x, that would be f(5) = 2*(5) - 3. Let me calculate that. 2 times 5 is 10, and then subtract 3, so 10 - 3 equals 7. So f(5) is 7. Got that part.

Now the next part is g(f(5) - 1). Wait, so I need to compute f(5) first, which I did, and then subtract 1 from that result. So f(5) is 7, so 7 minus 1 is 6. So now I have to find g(6), right? Because the expression inside the g function is f(5) - 1, which is 6.

But let me make sure I'm doing this correctly. The original expression is g(f(5) - 1). Is that the same as g(f(5) - 1)? Yes, that's how it's written. So first compute f(5), subtract 1, then apply g to that result.

So, as I said, f(5) is 7. Then 7 - 1 is 6. Then g(6) is... Well, g(x) is x + 1. So plugging in 6, that would be 6 + 1, which is 7. Therefore, the value of g(f(5) - 1) is 7? Wait, that seems straightforward, but let me check again to make sure I didn't skip any steps or make a mistake.

Let me go through it again step by step. First, compute f(5). f(x) = 2x - 3. So 2 times 5 is 10, minus 3 is 7. Correct. Then subtract 1 from that result: 7 - 1 = 6. Then apply g to 6. Since g(x) = x + 1, then g(6) = 6 + 1 = 7. So yes, that seems to check out.

But maybe I should consider if there's another way to interpret the original expression? For example, could it be g(f(5 - 1))? But no, the original problem says g(f(5) - 1). The parentheses are important here. The function g is being applied to the result of f(5) minus 1, not to f(5 - 1). So the order of operations is important here.

Let me just verify that. If it were g(f(5 - 1)), then we would first compute 5 - 1 = 4, then compute f(4), which would be 2*4 - 3 = 8 - 3 = 5, then apply g to 5, which would be 5 + 1 = 6. But that's a different answer. However, the original problem is g(f(5) - 1), so the parentheses are around f(5), meaning we compute f(5) first, then subtract 1. So my original calculation is correct.

Another way to think about it is that the expression is g evaluated at (f(5) - 1). So the inner function is f(5), then subtract 1, then plug into g.

Alternatively, maybe I can write out the composition of functions? Let me see. If I wanted to write g(f(x) - 1), that would be different, but in this case, it's specifically g(f(5) - 1). So no, it's not a composition of functions, it's just evaluating f at 5, then doing some operations on that result before plugging into g.

So, just to recap:

1. Calculate f(5):
   f(5) = 2*(5) - 3 = 10 - 3 = 7.

2. Subtract 1 from f(5):
   7 - 1 = 6.

3. Calculate g(6):
   g(6) = 6 + 1 = 7.

Therefore, the value is 7.

Wait, but let me check if there's any chance that the problem is written differently. For example, maybe the original problem is g(f(5 - 1))? But no, the user wrote "g(f(5)-1)", which is g of (f(5) minus 1). So the parentheses are only around the f(5).

Alternatively, maybe there's a typo in my thinking? Let me check once more.

Original problem: What is the value of g(f(5) - 1)?

So, f(5) is 7. Then subtract 1: 7 - 1 = 6. Then g(6) is 6 + 1 = 7.

Alternatively, if someone misreads the problem as g(f(5 - 1)), they would get a different answer, but that's not what's written here.

Alternatively, maybe the problem is written with different parentheses? Let me check the original problem again.

The user wrote: "What is the value of g(f(5)-1)?" So, the expression is g(f(5) - 1). So, the f(5) is inside the parentheses for the function g, but then subtract 1? Wait, no. Wait, hold on. Wait, is the expression g(f(5) - 1) or is it g(f(5 - 1))?

Wait, the way it's written is g(f(5)-1). So, according to order of operations, the f(5) is computed first, then subtract 1, then apply g. So, the expression is g( [f(5)] - 1 ).

But if it were g(f(5 - 1)), that would be different. But since there's no parentheses around 5 - 1, it's only f(5) minus 1.

Alternatively, maybe the user made a typo? But assuming the problem is written correctly, then my answer is 7.

Alternatively, maybe I need to check if the functions are defined differently? Let me check again.

The problem says f(x) = 2x - 3 and g(x) = x + 1. So f is a linear function with slope 2 and y-intercept -3, and g is a linear function with slope 1 and y-intercept 1.

So, f(5) is 2*5 - 3 = 10 - 3 = 7. Then subtract 1: 7 - 1 = 6. Then g(6) is 6 + 1 = 7.

Alternatively, maybe I can think of it as a composition of functions? For example, if we had g(f(x)) - 1, that would be different. Let me see. If it were g(f(x)) - 1, then that would be [f(x) + 1] - 1 = f(x). But that's not the case here.

Alternatively, maybe the problem is written as g(f(5 - 1))? Which would be g(f(4)) = g(2*4 - 3) = g(5) = 5 + 1 = 6. But again, that's different.

But since the problem says g(f(5) - 1), I think my original answer is correct.

Alternatively, maybe I can check with substitution. Let me define f and g as functions and compute step by step.

Let me write f(x) = 2x - 3. So f(5) is 2*5 - 3 = 7. Then f(5) - 1 is 7 - 1 = 6. Then g(6) is 6 + 1 = 7.

Alternatively, if I wanted to write this as a single expression, maybe I can substitute. Let me see.

g(f(5) - 1) = g( (2*5 - 3) - 1 ) = g( (10 - 3) - 1 ) = g(7 - 1) = g(6) = 6 + 1 = 7.

Yes, that's the same result.

Alternatively, maybe I can think of it as a composite function. If I wanted to write a function h(x) = g(f(x) - 1), then h(5) would be g(f(5) - 1). But that's exactly what the problem is asking. So h(x) = g(f(x) - 1) = (f(x) - 1) + 1 = f(x). Wait, that's interesting.

Wait, if h(x) = g(f(x) - 1), then substituting the functions, h(x) = [f(x) - 1] + 1 = f(x). So h(x) is just f(x). Therefore, h(5) = f(5) = 7.

Wait, so that's another way to see it. If you have g(f(x) - 1), since g is x + 1, then substituting, it's (f(x) - 1) + 1 = f(x). Therefore, the entire expression simplifies to f(x). Therefore, h(5) = f(5) = 7.

So that's another way to think about it. Therefore, regardless of the method, the answer is 7.

But let me check that again. If h(x) = g(f(x) - 1), then h(x) = g(f(x) - 1) = [f(x) - 1] + 1 = f(x). So yes, because g is adding 1, and we have f(x) - 1 inside, so adding 1 cancels out the -1. Therefore, h(x) = f(x). Therefore, h(5) = f(5) = 7.

Therefore, even if we think of it as a composite function, we get the same answer.

So, that seems to confirm that the answer is indeed 7.

But just to make sure, let me try plugging in numbers again.

Original problem: g(f(5) - 1).

Compute f(5): 2*5 - 3 = 10 - 3 = 7.

Subtract 1: 7 - 1 = 6.

Compute g(6): 6 + 1 = 7.

Yes, same result.

Alternatively, if I made a mistake in arithmetic? Let me check again.

2*5 is 10. 10 - 3 is 7. 7 - 1 is 6. 6 + 1 is 7.

No arithmetic errors here.

Alternatively, maybe the problem is written in a different way? For example, if it was g(f(5 - 1)), then 5 - 1 is 4, f(4) is 2*4 - 3 = 8 - 3 = 5, then g(5) is 5 + 1 = 6. But that's different. However, the problem says g(f(5) - 1), not g(f(5 - 1)).

Therefore, unless there's a misinterpretation of the parentheses, which I don't think so, the answer is 7.

Alternatively, maybe the problem is written with different grouping? For example, if it was g(f(5 - 1)), but the original problem is written as g(f(5)-1).

In the original problem, the user wrote: "g(f(5)-1)". So, the parentheses are only around the f(5). So, the expression is g( [f(5)] - 1 ).

Therefore, I think my answer is correct.

Alternatively, maybe the problem is from a source where there's a different notation? For example, sometimes people use different notations, but in standard math notation, parentheses are for function arguments. So, f(5) is the function f evaluated at 5, and then subtract 1.

Therefore, I think I can be confident that the answer is 7.

But just to make sure, let me think if there's any other possible way. For example, if someone thought that f(5) - 1 is f(5 - 1), but that would be incorrect.

In general, f(a) - b is not the same as f(a - b). Unless specified otherwise, they are different.

So, for example, f(5) - 1 is 7 - 1 = 6, whereas f(5 - 1) is f(4) = 5.

Therefore, unless there's a specific reason to think that the expression is f(5 - 1), which there isn't, the answer is 7.

Therefore, after all these checks, I think the answer is 7.

**Final Answer**
\boxed{7}

To evaluate the expression $ g(f(5) - 1) $, we proceed step by step using the given functions:

---

### Step 1: Evaluate $ f(5) $

We are given the function:

$$
f(x) = 2x - 3
$$

Substitute $ x = 5 $:

$$
f(5) = 2(5) - 3 = 10 - 3 = 7
$$

---

### Step 2: Compute $ f(5) - 1 $

Now subtract 1 from the result of $ f(5) $:

$$
f(5) - 1 = 7 - 1 = 6
$$

---

### Step 3: Evaluate $ g(6) $

We are given the function:

$$
g(x) = x + 1
$$

Substitute $ x = 6 $:

$$
g(6) = 6 + 1 = 7
$$

---

### Final Answer

$$
g(f(5) - 1) = \boxed{7}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of g(f(5) - 1). Let me start by recalling what the functions f(x) and g(x) are. The problem says f(x) is 2x minus 3, and g(x) is x plus 1. Alright, so first things first, I need to figure out what f(5) is.

Let me write that down. f(x) = 2x - 3. So if I plug in 5 for x, that would be f(5) = 2*(5) - 3. Let me calculate that. 2 times 5 is 10, and then subtract 3, so 10 - 3 equals 7. So f(5) is 7. Got that part.

Now the next part is g(f(5) - 1). Wait, so I need to compute f(5) first, which I did, and then subtract 1 from that result. So f(5) is 7, so 7 minus 1 is 6. So now I have to find g(6), right? Because the expression inside the g function is f(5) - 1, which is 6.

But let me make sure I'm doing this correctly. The original expression is g(f(5) - 1). Is that the same as g(f(5) - 1)? Yes, that's how it's written. So first compute f(5), subtract 1, then apply g to that result.

So, as I said, f(5) is 7. Then 7 - 1 is 6. Then g(6) is... Well, g(x) is x + 1. So plugging in 6, that would be 6 + 1, which is 7. Therefore, the value of g(f(5) - 1) is 7? Wait, that seems straightforward, but let me check again to make sure I didn't skip any steps or make a mistake.

Let me go through it again step by step. First, compute f(5). f(x) = 2x - 3. So 2 times 5 is 10, minus 3 is 7. Correct. Then subtract 1 from that result: 7 - 1 = 6. Then apply g to 6. Since g(x) = x + 1, then g(6) = 6 + 1 = 7. So yes, that seems to check out.

But maybe I should consider if there's another way to interpret the original expression? For example, could it be g(f(5 - 1))? But no, the original problem says g(f(5) - 1). The parentheses are important here. The function g is being applied to the result of f(5) minus 1, not to f(5 - 1). So the order of operations is important here.

Let me just verify that. If it were g(f(5 - 1)), then we would first compute 5 - 1 = 4, then compute f(4), which would be 2*4 - 3 = 8 - 3 = 5, then apply g to 5, which would be 5 + 1 = 6. But that's a different answer. However, the original problem is g(f(5) - 1), so the parentheses are around f(5), meaning we compute f(5) first, then subtract 1. So my original calculation is correct.

Another way to think about it is that the expression is g evaluated at (f(5) - 1). So the inner function is f(5), then subtract 1, then plug into g.

Alternatively, maybe I can write out the composition of functions? Let me see. If I wanted to write g(f(x) - 1), that would be different, but in this case, it's specifically g(f(5) - 1). So no, it's not a composition of functions, it's just evaluating f at 5, then doing some operations on that result before plugging into g.

So, just to recap:

1. Calculate f(5):
   f(5) = 2*(5) - 3 = 10 - 3 = 7.

2. Subtract 1 from f(5):
   7 - 1 = 6.

3. Calculate g(6):
   g(6) = 6 + 1 = 7.

Therefore, the value is 7.

Wait, but let me check if there's any chance that the problem is written differently. For example, maybe the original problem is g(f(5 - 1))? But no, the user wrote "g(f(5)-1)", which is g of (f(5) minus 1). So the parentheses are only around the f(5).

Alternatively, maybe there's a typo in my thinking? Let me check once more.

Original problem: What is the value of g(f(5) - 1)?

So, f(5) is 7. Then subtract 1: 7 - 1 = 6. Then g(6) is 6 + 1 = 7.

Alternatively, if someone misreads the problem as g(f(5 - 1)), they would get a different answer, but that's not what's written here.

Alternatively, maybe the problem is written with different parentheses? Let me check the original problem again.

The user wrote: "What is the value of g(f(5)-1)?" So, the expression is g(f(5) - 1). So, the f(5) is inside the parentheses for the function g, but then subtract 1? Wait, no. Wait, hold on. Wait, is the expression g(f(5) - 1) or is it g(f(5 - 1))?

Wait, the way it's written is g(f(5)-1). So, according to order of operations, the f(5) is computed first, then subtract 1, then apply g. So, the expression is g( [f(5)] - 1 ).

But if it were g(f(5 - 1)), that would be different. But since there's no parentheses around 5 - 1, it's only f(5) minus 1.

Alternatively, maybe the user made a typo? But assuming the problem is written correctly, then my answer is 7.

Alternatively, maybe I need to check if the functions are defined differently? Let me check again.

The problem says f(x) = 2x - 3 and g(x) = x + 1. So f is a linear function with slope 2 and y-intercept -3, and g is a linear function with slope 1 and y-intercept 1.

So, f(5) is 2*5 - 3 = 10 - 3 = 7. Then subtract 1: 7 - 1 = 6. Then g(6) is 6 + 1 = 7.

Alternatively, maybe I can think of it as a composition of functions? For example, if we had g(f(x)) - 1, that would be different. Let me see. If it were g(f(x)) - 1, then that would be [f(x) + 1] - 1 = f(x). But that's not the case here.

Alternatively, maybe the problem is written as g(f(5 - 1))? Which would be g(f(4)) = g(2*4 - 3) = g(5) = 5 + 1 = 6. But again, that's different.

But since the problem says g(f(5) - 1), I think my original answer is correct.

Alternatively, maybe I can check with substitution. Let me define f and g as functions and compute step by step.

Let me write f(x) = 2x - 3. So f(5) is 2*5 - 3 = 7. Then f(5) - 1 is 7 - 1 = 6. Then g(6) is 6 + 1 = 7.

Alternatively, if I wanted to write this as a single expression, maybe I can substitute. Let me see.

g(f(5) - 1) = g( (2*5 - 3) - 1 ) = g( (10 - 3) - 1 ) = g(7 - 1) = g(6) = 6 + 1 = 7.

Yes, that's the same result.

Alternatively, maybe I can think of it as a composite function. If I wanted to write a function h(x) = g(f(x) - 1), then h(5) would be g(f(5) - 1). But that's exactly what the problem is asking. So h(x) = g(f(x) - 1) = (f(x) - 1) + 1 = f(x). Wait, that's interesting.

Wait, if h(x) = g(f(x) - 1), then substituting the functions, h(x) = [f(x) - 1] + 1 = f(x). So h(x) is just f(x). Therefore, h(5) = f(5) = 7.

Wait, so that's another way to see it. If you have g(f(x) - 1), since g is x + 1, then substituting, it's (f(x) - 1) + 1 = f(x). Therefore, the entire expression simplifies to f(x). Therefore, h(5) = f(5) = 7.

So that's another way to think about it. Therefore, regardless of the method, the answer is 7.

But let me check that again. If h(x) = g(f(x) - 1), then h(x) = g(f(x) - 1) = [f(x) - 1] + 1 = f(x). So yes, because g is adding 1, and we have f(x) - 1 inside, so adding 1 cancels out the -1. Therefore, h(x) = f(x). Therefore, h(5) = f(5) = 7.

Therefore, even if we think of it as a composite function, we get the same answer.

So, that seems to confirm that the answer is indeed 7.

But just to make sure, let me try plugging in numbers again.

Original problem: g(f(5) - 1).

Compute f(5): 2*5 - 3 = 10 - 3 = 7.

Subtract 1: 7 - 1 = 6.

Compute g(6): 6 + 1 = 7.

Yes, same result.

Alternatively, if I made a mistake in arithmetic? Let me check again.

2*5 is 10. 10 - 3 is 7. 7 - 1 is 6. 6 + 1 is 7.

No arithmetic errors here.

Alternatively, maybe the problem is written in a different way? For example, if it was g(f(5 - 1)), then 5 - 1 is 4, f(4) is 2*4 - 3 = 8 - 3 = 5, then g(5) is 5 + 1 = 6. But that's different. However, the problem says g(f(5) - 1), not g(f(5 - 1)).

Therefore, unless there's a misinterpretation of the parentheses, which I don't think so, the answer is 7.

Alternatively, maybe the problem is written with different grouping? For example, if it was g(f(5 - 1)), but the original problem is written as g(f(5)-1).

In the original problem, the user wrote: "g(f(5)-1)". So, the parentheses are only around the f(5). So, the expression is g( [f(5)] - 1 ).

Therefore, I think my answer is correct.

Alternatively, maybe the problem is from a source where there's a different notation? For example, sometimes people use different notations, but in standard math notation, parentheses are for function arguments. So, f(5) is the function f evaluated at 5, and then subtract 1.

Therefore, I think I can be confident that the answer is 7.

But just to make sure, let me think if there's any other possible way. For example, if someone thought that f(5) - 1 is f(5 - 1), but that would be incorrect.

In general, f(a) - b is not the same as f(a - b). Unless specified otherwise, they are different.

So, for example, f(5) - 1 is 7 - 1 = 6, whereas f(5 - 1) is f(4) = 5.

Therefore, unless there's a specific reason to think that the expression is f(5 - 1), which there isn't, the answer is 7.

Therefore, after all these checks, I think the answer is 7.

**Final Answer**
\boxed{7}

To evaluate the expression $ g(f(5) - 1) $, we proceed step by step using the given functions:

---

### Step 1: Evaluate $ f(5) $

We are given the function:

$$
f(x) = 2x - 3
$$

Substitute $ x = 5 $:

$$
f(5) = 2(5) - 3 = 10 - 3 = 7
$$

---

### Step 2: Compute $ f(5) - 1 $

Now subtract 1 from the result of $ f(5) $:

$$
f(5) - 1 = 7 - 1 = 6
$$

---

### Step 3: Evaluate $ g(6) $

We are given the function:

$$
g(x) = x + 1
$$

Substitute $ x = 6 $:

$$
g(6) = 6 + 1 = 7
$$

---

### Final Answer

$$
g(f(5) - 1) = \boxed{7}
$$

Solution: We have $f(5) = 2(5) -3 = 7$, so $g(f(5)-1) = g(7-1) = g(6) = 6+1 = \boxed{7}$.