Explanation:
Okay, so I need to figure out how many ways the Senate committee can sit around a circular table with the condition that all members of each party sit next to each other. The committee has 5 Democrats, 5 Republicans, and 1 Independent. And rotations are considered the same seating. Hmm, let me think.

First, circular permutations can be tricky because rotations are considered the same. Normally, for n people around a circular table, the number of arrangements is (n-1)! because rotating the whole table doesn't create a new arrangement. But here, there are additional constraints: each party must sit together. So maybe I can treat each party as a single block first?

Wait, but there are three parties: Democrats, Republicans, and Independent. However, the Independent is only one person. So maybe I need to consider the blocks. Let me think.

If all members of each party must sit together, then we can think of the Democrats as one block, the Republicans as another block, and the Independent as a single block. But since the Independent is just one person, that block is just one person. However, the problem is that the Independent is a single person, so maybe they can be considered as a block of size 1. But how does that affect the arrangement?

So, if we have three blocks: D (Democrats), R (Republicans), and I (Independent). Since the table is circular, arranging these blocks around the table. Normally, arranging k blocks around a circular table would be (k-1)! ways. But here, k is 3, so (3-1)! = 2 ways? Wait, but then we also have to consider the internal arrangements within each block.

But hold on, the Democrats are 5 people, so within the D block, they can be arranged in 5! ways. Similarly, the Republicans are 5 people, so 5! ways. The Independent is just 1 person, so only 1 way. So total arrangements would be (number of ways to arrange the blocks) multiplied by (internal arrangements of each block). So that would be 2 * 5! * 5! * 1. But wait, is that correct?

Wait, but maybe I need to check if the blocks are distinguishable. Since the Independent is a single person, but the other two are blocks of 5. So the blocks are distinguishable because they are different parties. So arranging them around the table. So yes, the number of ways to arrange the blocks is (3-1)! = 2. Then multiply by the internal arrangements. So total is 2 * 5! * 5!.

But wait, let me think again. Suppose we fix one block to account for the circular table. For example, fix the Democrats block in a position. Then arrange the other two blocks (Republicans and Independent) around the table. Since it's circular, fixing one block removes rotational symmetry. Then the number of ways to arrange the remaining blocks is 2! (since there are two blocks left: Republicans and Independent). Then multiply by internal arrangements. So that would be 2! * 5! * 5! * 1. Which is 2 * 5! * 5!.

But earlier I thought (k-1)! where k=3 is 2, which is same as 2! So that seems consistent. So total number of arrangements is 2 * 5! * 5!.

But wait, but the Independent is a single person. So maybe there's another way to think about it. Let me check.

Alternatively, suppose we consider the entire arrangement. Since all Democrats are together, all Republicans are together, and the Independent is alone. So the seating is divided into three contiguous blocks: D, R, I. But since it's a circular table, the order of these blocks can be arranged in different ways. However, because it's circular, the number of distinct arrangements is (number of permutations of the blocks) divided by the number of rotations. Wait, but if we fix one block, then the rest can be arranged in 2! ways. So that's the same as before.

Alternatively, if we didn't fix any block, the number of circular permutations of the blocks is (3-1)! = 2. So same result. So then multiply by internal arrangements. So 2 * 5! * 5!.

But wait, but the Independent is a single person. So maybe there's something else? Let me think. Suppose we have three blocks: D, R, I. Each block is a group. The D block has 5 people, R block has 5 people, I block has 1 person. So arranging these three blocks around the table. Since it's circular, the number of ways to arrange the blocks is (3-1)! = 2. Then, for each block, we can permute the people inside. So D block: 5! ways, R block: 5! ways, I block: 1! way. So total is 2 * 5! * 5! * 1! = 2 * 5! * 5!.

But wait, is that all? Let me check with a simpler case. Suppose there are two parties: 1 Democrat and 1 Republican. How many ways? If they have to sit together, but since they are two blocks. Wait, but if it's a circular table, then arranging two blocks around the table. Since it's circular, there's only 1 way to arrange two blocks (since rotating doesn't change the arrangement). But if we fix one block, then the other can be arranged in 1! ways. So total arrangements would be 1! * 1! * 1! = 1? But actually, if you have two people, one Democrat and one Republican, sitting around a circular table, there's only 1 way because rotations are considered the same. But if they are sitting in a line, it would be 2 ways. But since it's circular, it's 1. But according to the formula, (k-1)! where k=2 is 1, multiplied by internal arrangements (1! * 1!) = 1. So that works.

Another example: 2 Democrats and 1 Republican. So blocks: D (2 people) and R (1 person). Number of circular arrangements: (2-1)! = 1. Multiply by internal arrangements: 2! * 1! = 2. So total 2. Let's check. If we fix the D block, then the R can be to the left or right? Wait, but in circular table, fixing the D block, the R can be placed in only one position? Wait, no. Wait, if you fix the D block, then the R block is adjacent to the D block. But since it's a circular table, once you fix the D block, the R block can be placed in only one position? Wait, but if there are two blocks, then once you fix one block, the other block is fixed in position. So the number of arrangements is 1. But internal arrangements: 2! for D and 1! for R. So total 2. But if you have two blocks, D and R, around a circular table, then there's only one way to arrange the blocks, but the D block can be arranged in 2! ways and R in 1! way. So total 2. Which makes sense. For example, if you have two people in D: A and B, and one person R: C. Fixing the D block as A-B, then the R can be to the left or right? Wait, but if you fix the D block, then the R block is adjacent to the D block. But since it's a circular table, once you fix the D block, the R block can be placed in only one position? Wait, maybe not. Wait, if you fix the D block, then the R block can be placed either clockwise or counter-clockwise? Wait, but if you fix the D block, then the entire table is fixed? Wait, maybe not. Let me think.

Suppose we fix the D block in a specific position. Let's say the D block is sitting from position 1 to 2. Then the R block can be either to the left of position 1 or to the right of position 2. But since it's a circular table, left of position 1 is the same as right of position 2? Wait, no. If you fix the D block, then the R block can be placed in only one position? Wait, no. If you fix the D block, then the R block can be placed in two different positions? Wait, but if you fix the D block, then the entire arrangement is fixed? Wait, maybe not. Let me think of it as arranging the blocks. If there are two blocks, D and R, then in a circular table, there's only one way to arrange them, but the D block can be rotated. Wait, but we fixed the D block. So if we fix the D block, then the R block is fixed in position. So the number of arrangements is 1. But if we don't fix the D block, then the number of arrangements is (2-1)! = 1. So that seems to be consistent. So in that case, the total number of arrangements is 1 * 2! * 1! = 2. Which would correspond to the two different arrangements of the D block: A-B or B-A, with R fixed in the other position. So that seems correct.

Therefore, returning to the original problem, if we have three blocks: D, R, I. Then the number of circular arrangements is (3-1)! = 2. Multiply by internal arrangements: 5! for D, 5! for R, 1! for I. So total is 2 * 5! * 5!.

But wait, but the Independent is a single person. So maybe there's another consideration? For example, if the Independent is considered as a block, but since they are only one person, does that affect anything? I don't think so. Because even though they are a single person, they are still a block, so arranging the blocks is the same as arranging the three blocks. So the calculation seems okay.

But let me think again. Suppose we have three blocks: D, R, I. The number of ways to arrange them around the table is (3-1)! = 2. Then, for each of these arrangements, we can permute the members within each block. So D: 5!, R: 5!, I: 1. So total is 2 * 5! * 5!.

But let me check with another approach. Let's imagine "breaking" the circular table into a line. For circular permutations, sometimes it's helpful to fix one position to convert it into a linear permutation. So if we fix the position of the D block, then we can arrange the other blocks linearly. So fixing the D block, then the remaining blocks are R and I. These can be arranged in 2! ways. Then, within the D block, 5! ways, within R block, 5! ways, and within I block, 1 way. So total is 2! * 5! * 5! = 2 * 5! * 5!.

Same result. So that seems to confirm it. So maybe the answer is 2 * 5! * 5!.

But wait, let me check if there's a different way to think about it. For example, suppose we consider the entire seating. Since all Democrats are together, all Republicans are together, and the Independent is alone. So the seating is a sequence of D block, R block, I block, or any permutation of these blocks. But since it's a circular table, the number of distinct permutations is (number of linear permutations)/number of rotations. Wait, but how does that work?

In linear arrangements, the number of ways to arrange the blocks would be 3! = 6. But since it's circular, we divide by the number of rotations, which is 3 (since there are 3 blocks). Wait, but that would give 6 / 3 = 2, which matches the previous result. So then, the number of circular arrangements of the blocks is 2. Then multiply by internal arrangements: 5! * 5! * 1. So total is 2 * 5! * 5!.

Therefore, that seems consistent. So maybe the answer is 2 * 5! * 5!.

But let me check if there's something missing here. For example, the Independent is a single person. Is there a possibility that the Independent could be between two blocks? Wait, but since all the other blocks are together, the Independent is a single person, so they have to be between two blocks. But since the blocks are arranged in a circle, the Independent is just one person, so the arrangement is D block, R block, I, or D block, I, R block, or R block, D block, I, etc. But since it's circular, some of these are equivalent.

But we already considered that by using (k-1)! for the blocks. So I think that's accounted for.

Alternatively, maybe I need to think of the Independent as part of the blocks. Wait, but the problem says "all the members of each party all sit next to each other". So the Independent is a party of one. So they have to sit next to each other, which they trivially do because they are alone. So that's okay.

Wait, but maybe there's a different interpretation. Suppose the Independent is not considered a party? Wait, the problem says "all the members of each party all sit next to each other". The parties are Democrats, Republicans, and Independent. So the Independent is a party with one member. So they have to sit together, which they do. So that's okay.

Therefore, I think the calculation is correct. So the answer is 2 * 5! * 5!.

But let me compute the numerical value to check. 5! is 120, so 120 * 120 = 14,400. Multiply by 2 is 28,800. So 28,800 ways.

But let me check again if there's a different approach. For example, suppose we first consider the entire group. Since all Democrats are together, all Republicans are together, and the Independent is alone. So imagine that we have three blocks: D, R, I. The number of ways to arrange these blocks around the table is (3-1)! = 2. Then, for each block, we can permute the members. So D: 5!, R: 5!, I: 1. So total is 2 * 5! * 5!.

Alternatively, if we didn't consider the circular table, and just thought of it as linear, then the number would be 3! * 5! * 5! * 1. But since it's circular, we divide by 3 (the number of rotations), so 3! / 3 = 2, which gives the same result. So that seems to confirm again.

But let me think if there's a possibility that the Independent could be considered as part of another block? For example, maybe the problem is that the Independent is not a party? Wait, the problem says "all the members of each party all sit next to each other". The parties are Democrats, Republicans, and Independent. So the Independent is a party. Therefore, they must sit together. So the Independent is a single person, so they are sitting together by themselves. So that's okay.

Alternatively, maybe the problem is that the Independent is not a party? But the problem says "a Senate committee has 5 Democrats, 5 Republicans, and 1 Independent". So the Independent is a member, but not part of a party? Wait, but the problem says "all the members of each party all sit next to each other". So if the Independent is not part of a party, then they don't have to sit together? Wait, but the problem says "all the members of each party all sit next to each other". So the parties are Democrats, Republicans, and the Independent? Or is the Independent not considered a party?

Wait, this is a critical point. The problem says "all the members of each party all sit next to each other". The committee has 5 Democrats, 5 Republicans, and 1 Independent. So the question is: is the Independent considered a party? If so, then they must sit together. If not, then they don't have to. But the problem says "each party", so the parties are Democrats, Republicans, and the Independent? Or is the Independent not a party?

This is ambiguous. Wait, but in the Senate, the Independent is not a party. So maybe the problem is that the Independent is not a party, so they don't have to sit together. But the problem says "all the members of each party all sit next to each other". So if the Independent is not a party, then they don't have to sit together. But the problem says "all the members of each party all sit next to each other". So maybe the parties are only Democrats and Republicans? But the problem mentions the Independent as a member. Wait, the problem says "a Senate committee has 5 Democrats, 5 Republicans, and 1 Independent". So maybe the parties are Democrats, Republicans, and the Independent is not a party. Therefore, the condition is that all Democrats sit together, all Republicans sit together, and the Independent can be anywhere. But the problem says "all the members of each party all sit next to each other". So if the Independent is not a party, then they don't have to sit together. But the problem says "each party", so maybe the Independent is considered a party? Or maybe the problem is that the Independent is a party? Hmm.

Wait, the problem says "all the members of each party all sit next to each other". So if there are three parties: Democrats, Republicans, and Independent. Then each party's members must sit together. Therefore, the Independent, being a party of one, must sit together. Which is trivial. Therefore, the answer is as I calculated before: 2 * 5! * 5!.

But maybe the problem is that the Independent is not a party, so only Democrats and Republicans need to sit together. Then the problem would be different. But the problem says "each party", so if there are three parties, then the Independent is a party. But maybe the problem is that the Independent is not a party. Wait, the problem is from a math competition or something? Maybe the standard way is that the Independent is not considered a party. But the problem says "each party", so maybe the parties are Democrats and Republicans, and the Independent is not a party. Therefore, the condition is that Democrats sit together, Republicans sit together, and the Independent can be anywhere. But then the problem would have said "each of the two parties" or something. But the problem says "each party", and there are three parties. Wait, the problem says "a Senate committee has 5 Democrats, 5 Republicans, and 1 Independent". So maybe the parties are Democrats, Republicans, and the Independent is not a party. Therefore, the problem is that Democrats must sit together, Republicans must sit together, and the Independent can be anywhere. But then the problem would have said "each of the two parties" or "the two parties". But the problem says "each party", so maybe the Independent is considered a party. This is a critical ambiguity.

Wait, let me check the exact wording: "all the members of each party all sit next to each other". The committee has 5 Democrats, 5 Republicans, and 1 Independent. So the parties are Democrats, Republicans, and the Independent? Or is the Independent not a party? In real life, the Independent is not a party, but in the context of the problem, maybe they consider the Independent as a party. But the problem says "each party", so if there are three parties, then the answer is as I calculated. If there are two parties, then the answer is different.

But since the problem mentions the Independent as a separate member, maybe they are not considered a party. Therefore, the problem is that Democrats must sit together, Republicans must sit together, and the Independent can be anywhere. But then the problem would have said "each of the two parties" or "the two parties". However, the problem says "each party", so maybe the Independent is considered a party. Therefore, the answer is 2 * 5! * 5!.

But to resolve this ambiguity, let me think again. If the Independent is not a party, then the problem is that Democrats and Republicans must sit together, and the Independent can be anywhere. So how would that affect the calculation?

In that case, we have two blocks: Democrats and Republicans, and the Independent. So the total number of people is 5 + 5 + 1 = 11. But the two blocks (Democrats and Republicans) must sit together, and the Independent is separate. So how to arrange them around the table?

Wait, if the Democrats and Republicans are each blocks, and the Independent is a single person. So we have three "units": D block, R block, and I. So same as before. Therefore, the number of circular arrangements is (3-1)! = 2, multiplied by internal arrangements: 5! * 5! * 1. So same answer. Therefore, regardless of whether the Independent is considered a party or not, the answer is the same. Because even if the Independent is not a party, they are treated as a single unit. Therefore, the answer is 2 * 5! * 5!.

Therefore, maybe the answer is indeed 2 * 5! * 5!.

But let me check again. Suppose the problem had said that only Democrats and Republicans need to sit together, and the Independent can be anywhere. Then, how would that affect the calculation?

In that case, we have two blocks: D and R, and the Independent. So arranging three units around the table: D, R, I. So same as before. So the number of circular arrangements is (3-1)! = 2. Multiply by internal arrangements: 5! * 5! * 1. So same answer. Therefore, regardless of whether the Independent is considered a party or not, the answer is same. Therefore, the answer is 2 * 5! * 5!.

Therefore, I think that is the answer.

But let me check with another approach. Suppose we first consider the Democrats as a block, Republicans as a block, and the Independent as a single person. So total of 3 units. The number of circular arrangements is (3-1)! = 2. Then, for each arrangement, we can permute the Democrats in 5! ways, Republicans in 5! ways, and the Independent in 1 way. So total is 2 * 5! * 5!.

Alternatively, if we think of the Independent as part of the Democrats or Republicans? No, because they are separate. So that's not the case.

Alternatively, suppose we first arrange the Democrats and Republicans as blocks, and then place the Independent. But since it's a circular table, arranging the blocks and the Independent.

But again, same as before.

Alternatively, think of the entire seating. Since the Democrats are together, they form a block. Similarly, the Republicans form a block. The Independent is alone. So the seating is a circular arrangement of these three blocks. So the number of ways is (number of circular permutations of the blocks) * (internal permutations). Which is 2 * 5! * 5!.

Therefore, I think this is correct.

But let me check with numbers. Suppose we have smaller numbers. Let's say 1 Democrat, 1 Republican, and 1 Independent. How many ways?

If all must sit together, but since each party is a single person. So the number of circular arrangements. Since there are three blocks (each of size 1). The number of circular arrangements is (3-1)! = 2. Multiply by internal arrangements: 1! * 1! * 1! = 1. So total 2. Which is correct. Because in a circular table with three people, there are (3-1)! = 2 arrangements. So that works.

Another example: 2 Democrats, 1 Republican, 1 Independent. So blocks: D (2), R (1), I (1). Number of circular arrangements: (3-1)! = 2. Multiply by internal arrangements: 2! * 1! * 1! = 2. Total 2 * 2 = 4. Let's check. Fix the D block. Then the other two blocks (R and I) can be arranged in 2! ways. So 2 ways. For each, the D block can be arranged in 2! ways. So total 2 * 2 = 4. Which is correct. For example, if the D block is AB, then R and I can be arranged as R-I or I-R. So two arrangements. And AB can be BA, so total 2*2=4. Which is correct. So that seems to check out.

Therefore, returning to the original problem, the answer should be 2 * 5! * 5!.

But let me check if there's a different interpretation where the Independent is not considered a block. For example, if the problem requires that Democrats sit together, Republicans sit together, and the Independent is not required to sit with anyone. Then, how would that affect the calculation?

In that case, we have two blocks: D and R, and the Independent. So arranging three units around the table: D, R, I. Which is same as before. So same answer. Therefore, regardless of whether the Independent is considered a party or not, the answer is same. Therefore, the answer is 2 * 5! * 5!.

Therefore, I think that is the answer.

But let me check once more. Suppose the problem had said that only Democrats and Republicans need to sit together, and the Independent can be anywhere. Then, the number of ways would be:

First, treat D and R as blocks. So two blocks. Then, the Independent is a single person. So arranging three units: D, R, I. Which is same as before. So (3-1)! = 2. Multiply by internal arrangements: 5! * 5! * 1. So same answer. Therefore, regardless of the Independent's status, the answer is same. Therefore, the answer is 2 * 5! * 5!.

Therefore, I think that is the answer. So the final answer is 2 * 5! * 5!.

Calculating that, 5! is 120, so 120 * 120 = 14,400. Multiply by 2: 28,800. So 28,800 ways.

But let me check if there's a different approach where the answer is different. For example, suppose we first consider the seating of the Democrats and Republicans as blocks, and then place the Independent.

But since the table is circular, we can fix the position of one block to eliminate rotational symmetry. Let's fix the Democrats block. Then, the Republicans block and the Independent can be arranged in 2! ways around the table. Then, the Democrats can be arranged in 5! ways, the Republicans in 5! ways, and the Independent in 1 way. So total is 2! * 5! * 5! = 2 * 120 * 120 = 28,800. Same result.

Alternatively, if we didn't fix the Democrats block, then the number of ways to arrange the blocks is (number of circular permutations) = (3-1)! = 2. Then same result.

Therefore, I think that is correct.

Therefore, the answer is 2 * 5! * 5! = 28,800. So boxed answer is \boxed{28800}.

**Final Answer**
\boxed{28800}

To solve the problem of determining how many ways the Senate committee can sit around a circular table with the condition that all members of each party sit next to each other, we proceed step by step.

---

### **Understanding the Problem**

We are given:

- **5 Democrats**
- **5 Republicans**
- **1 Independent**

The condition is that **all members of each party must sit next to each other**. This means:

- The **5 Democrats** must form a **block**.
- The **5 Republicans** must form another **block**.
- The **1 Independent** is a **single person**, so they form a **block of size 1**.

Thus, we are arranging **three blocks** around a **circular table**.

---

### **Step 1: Circular Permutations of Blocks**

In circular arrangements, the number of distinct ways to arrange **k** distinguishable blocks is:

$$
(k - 1)!
$$

Here, we have **3 blocks** (D, R, I), so the number of circular arrangements is:

$$
(3 - 1)! = 2
$$

---

### **Step 2: Internal Arrangements Within Each Block**

Each block can be internally permuted:

- **Democrats block**: 5 people → $5!$ ways
- **Republicans block**: 5 people → $5!$ ways
- **Independent block**: 1 person → $1!$ way

So, the total number of internal arrangements is:

$$
5! \times 5! \times 1! = 120 \times 120 \times 1 = 14,400
$$

---

### **Step 3: Total Number of Arrangements**

Now, multiply the number of circular arrangements of the blocks by the number of internal arrangements:

$$
\text{Total} = 2 \times 5! \times 5! = 2 \times 120 \times 120 = 28,800
$$

---

### **Final Answer**

$$
\boxed{28800}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many ways the Senate committee can sit around a circular table with the condition that all members of each party sit next to each other. The committee has 5 Democrats, 5 Republicans, and 1 Independent. And rotations are considered the same seating. Hmm, let me think.

First, circular permutations can be tricky because rotations are considered the same. Normally, for n people around a circular table, the number of arrangements is (n-1)! because rotating the whole table doesn't create a new arrangement. But here, there are additional constraints: each party must sit together. So maybe I can treat each party as a single block first?

Wait, but there are three parties: Democrats, Republicans, and Independent. However, the Independent is only one person. So maybe I need to consider the blocks. Let me think.

If all members of each party must sit together, then we can think of the Democrats as one block, the Republicans as another block, and the Independent as a single block. But since the Independent is just one person, that block is just one person. However, the problem is that the Independent is a single person, so maybe they can be considered as a block of size 1. But how does that affect the arrangement?

So, if we have three blocks: D (Democrats), R (Republicans), and I (Independent). Since the table is circular, arranging these blocks around the table. Normally, arranging k blocks around a circular table would be (k-1)! ways. But here, k is 3, so (3-1)! = 2 ways? Wait, but then we also have to consider the internal arrangements within each block.

But hold on, the Democrats are 5 people, so within the D block, they can be arranged in 5! ways. Similarly, the Republicans are 5 people, so 5! ways. The Independent is just 1 person, so only 1 way. So total arrangements would be (number of ways to arrange the blocks) multiplied by (internal arrangements of each block). So that would be 2 * 5! * 5! * 1. But wait, is that correct?

Wait, but maybe I need to check if the blocks are distinguishable. Since the Independent is a single person, but the other two are blocks of 5. So the blocks are distinguishable because they are different parties. So arranging them around the table. So yes, the number of ways to arrange the blocks is (3-1)! = 2. Then multiply by the internal arrangements. So total is 2 * 5! * 5!.

But wait, let me think again. Suppose we fix one block to account for the circular table. For example, fix the Democrats block in a position. Then arrange the other two blocks (Republicans and Independent) around the table. Since it's circular, fixing one block removes rotational symmetry. Then the number of ways to arrange the remaining blocks is 2! (since there are two blocks left: Republicans and Independent). Then multiply by internal arrangements. So that would be 2! * 5! * 5! * 1. Which is 2 * 5! * 5!.

But earlier I thought (k-1)! where k=3 is 2, which is same as 2! So that seems consistent. So total number of arrangements is 2 * 5! * 5!.

But wait, but the Independent is a single person. So maybe there's another way to think about it. Let me check.

Alternatively, suppose we consider the entire arrangement. Since all Democrats are together, all Republicans are together, and the Independent is alone. So the seating is divided into three contiguous blocks: D, R, I. But since it's a circular table, the order of these blocks can be arranged in different ways. However, because it's circular, the number of distinct arrangements is (number of permutations of the blocks) divided by the number of rotations. Wait, but if we fix one block, then the rest can be arranged in 2! ways. So that's the same as before.

Alternatively, if we didn't fix any block, the number of circular permutations of the blocks is (3-1)! = 2. So same result. So then multiply by internal arrangements. So 2 * 5! * 5!.

But wait, but the Independent is a single person. So maybe there's something else? Let me think. Suppose we have three blocks: D, R, I. Each block is a group. The D block has 5 people, R block has 5 people, I block has 1 person. So arranging these three blocks around the table. Since it's circular, the number of ways to arrange the blocks is (3-1)! = 2. Then, for each block, we can permute the people inside. So D block: 5! ways, R block: 5! ways, I block: 1! way. So total is 2 * 5! * 5! * 1! = 2 * 5! * 5!.

But wait, is that all? Let me check with a simpler case. Suppose there are two parties: 1 Democrat and 1 Republican. How many ways? If they have to sit together, but since they are two blocks. Wait, but if it's a circular table, then arranging two blocks around the table. Since it's circular, there's only 1 way to arrange two blocks (since rotating doesn't change the arrangement). But if we fix one block, then the other can be arranged in 1! ways. So total arrangements would be 1! * 1! * 1! = 1? But actually, if you have two people, one Democrat and one Republican, sitting around a circular table, there's only 1 way because rotations are considered the same. But if they are sitting in a line, it would be 2 ways. But since it's circular, it's 1. But according to the formula, (k-1)! where k=2 is 1, multiplied by internal arrangements (1! * 1!) = 1. So that works.

Another example: 2 Democrats and 1 Republican. So blocks: D (2 people) and R (1 person). Number of circular arrangements: (2-1)! = 1. Multiply by internal arrangements: 2! * 1! = 2. So total 2. Let's check. If we fix the D block, then the R can be to the left or right? Wait, but in circular table, fixing the D block, the R can be placed in only one position? Wait, no. Wait, if you fix the D block, then the R block is adjacent to the D block. But since it's a circular table, once you fix the D block, the R block can be placed in only one position? Wait, but if there are two blocks, then once you fix one block, the other block is fixed in position. So the number of arrangements is 1. But internal arrangements: 2! for D and 1! for R. So total 2. But if you have two blocks, D and R, around a circular table, then there's only one way to arrange the blocks, but the D block can be arranged in 2! ways and R in 1! way. So total 2. Which makes sense. For example, if you have two people in D: A and B, and one person R: C. Fixing the D block as A-B, then the R can be to the left or right? Wait, but if you fix the D block, then the R block is adjacent to the D block. But since it's a circular table, once you fix the D block, the R block can be placed in only one position? Wait, maybe not. Wait, if you fix the D block, then the R block can be placed either clockwise or counter-clockwise? Wait, but if you fix the D block, then the entire table is fixed? Wait, maybe not. Let me think.

Suppose we fix the D block in a specific position. Let's say the D block is sitting from position 1 to 2. Then the R block can be either to the left of position 1 or to the right of position 2. But since it's a circular table, left of position 1 is the same as right of position 2? Wait, no. If you fix the D block, then the R block can be placed in only one position? Wait, no. If you fix the D block, then the R block can be placed in two different positions? Wait, but if you fix the D block, then the entire arrangement is fixed? Wait, maybe not. Let me think of it as arranging the blocks. If there are two blocks, D and R, then in a circular table, there's only one way to arrange them, but the D block can be rotated. Wait, but we fixed the D block. So if we fix the D block, then the R block is fixed in position. So the number of arrangements is 1. But if we don't fix the D block, then the number of arrangements is (2-1)! = 1. So that seems to be consistent. So in that case, the total number of arrangements is 1 * 2! * 1! = 2. Which would correspond to the two different arrangements of the D block: A-B or B-A, with R fixed in the other position. So that seems correct.

Therefore, returning to the original problem, if we have three blocks: D, R, I. Then the number of circular arrangements is (3-1)! = 2. Multiply by internal arrangements: 5! for D, 5! for R, 1! for I. So total is 2 * 5! * 5!.

But wait, but the Independent is a single person. So maybe there's another consideration? For example, if the Independent is considered as a block, but since they are only one person, does that affect anything? I don't think so. Because even though they are a single person, they are still a block, so arranging the blocks is the same as arranging the three blocks. So the calculation seems okay.

But let me think again. Suppose we have three blocks: D, R, I. The number of ways to arrange them around the table is (3-1)! = 2. Then, for each of these arrangements, we can permute the members within each block. So D: 5!, R: 5!, I: 1. So total is 2 * 5! * 5!.

But let me check with another approach. Let's imagine "breaking" the circular table into a line. For circular permutations, sometimes it's helpful to fix one position to convert it into a linear permutation. So if we fix the position of the D block, then we can arrange the other blocks linearly. So fixing the D block, then the remaining blocks are R and I. These can be arranged in 2! ways. Then, within the D block, 5! ways, within R block, 5! ways, and within I block, 1 way. So total is 2! * 5! * 5! = 2 * 5! * 5!.

Same result. So that seems to confirm it. So maybe the answer is 2 * 5! * 5!.

But wait, let me check if there's a different way to think about it. For example, suppose we consider the entire seating. Since all Democrats are together, all Republicans are together, and the Independent is alone. So the seating is a sequence of D block, R block, I block, or any permutation of these blocks. But since it's a circular table, the number of distinct permutations is (number of linear permutations)/number of rotations. Wait, but how does that work?

In linear arrangements, the number of ways to arrange the blocks would be 3! = 6. But since it's circular, we divide by the number of rotations, which is 3 (since there are 3 blocks). Wait, but that would give 6 / 3 = 2, which matches the previous result. So then, the number of circular arrangements of the blocks is 2. Then multiply by internal arrangements: 5! * 5! * 1. So total is 2 * 5! * 5!.

Therefore, that seems consistent. So maybe the answer is 2 * 5! * 5!.

But let me check if there's something missing here. For example, the Independent is a single person. Is there a possibility that the Independent could be between two blocks? Wait, but since all the other blocks are together, the Independent is a single person, so they have to be between two blocks. But since the blocks are arranged in a circle, the Independent is just one person, so the arrangement is D block, R block, I, or D block, I, R block, or R block, D block, I, etc. But since it's circular, some of these are equivalent.

But we already considered that by using (k-1)! for the blocks. So I think that's accounted for.

Alternatively, maybe I need to think of the Independent as part of the blocks. Wait, but the problem says "all the members of each party all sit next to each other". So the Independent is a party of one. So they have to sit next to each other, which they trivially do because they are alone. So that's okay.

Wait, but maybe there's a different interpretation. Suppose the Independent is not considered a party? Wait, the problem says "all the members of each party all sit next to each other". The parties are Democrats, Republicans, and Independent. So the Independent is a party with one member. So they have to sit together, which they do. So that's okay.

Therefore, I think the calculation is correct. So the answer is 2 * 5! * 5!.

But let me compute the numerical value to check. 5! is 120, so 120 * 120 = 14,400. Multiply by 2 is 28,800. So 28,800 ways.

But let me check again if there's a different approach. For example, suppose we first consider the entire group. Since all Democrats are together, all Republicans are together, and the Independent is alone. So imagine that we have three blocks: D, R, I. The number of ways to arrange these blocks around the table is (3-1)! = 2. Then, for each block, we can permute the members. So D: 5!, R: 5!, I: 1. So total is 2 * 5! * 5!.

Alternatively, if we didn't consider the circular table, and just thought of it as linear, then the number would be 3! * 5! * 5! * 1. But since it's circular, we divide by 3 (the number of rotations), so 3! / 3 = 2, which gives the same result. So that seems to confirm again.

But let me think if there's a possibility that the Independent could be considered as part of another block? For example, maybe the problem is that the Independent is not a party? Wait, the problem says "all the members of each party all sit next to each other". The parties are Democrats, Republicans, and Independent. So the Independent is a party. Therefore, they must sit together. So the Independent is a single person, so they are sitting together by themselves. So that's okay.

Alternatively, maybe the problem is that the Independent is not a party? But the problem says "a Senate committee has 5 Democrats, 5 Republicans, and 1 Independent". So the Independent is a member, but not part of a party? Wait, but the problem says "all the members of each party all sit next to each other". So if the Independent is not part of a party, then they don't have to sit together? Wait, but the problem says "all the members of each party all sit next to each other". So the parties are Democrats, Republicans, and the Independent? Or is the Independent not considered a party?

Wait, this is a critical point. The problem says "all the members of each party all sit next to each other". The committee has 5 Democrats, 5 Republicans, and 1 Independent. So the question is: is the Independent considered a party? If so, then they must sit together. If not, then they don't have to. But the problem says "each party", so the parties are Democrats, Republicans, and the Independent? Or is the Independent not a party?

This is ambiguous. Wait, but in the Senate, the Independent is not a party. So maybe the problem is that the Independent is not a party, so they don't have to sit together. But the problem says "all the members of each party all sit next to each other". So if the Independent is not a party, then they don't have to sit together. But the problem says "all the members of each party all sit next to each other". So maybe the parties are only Democrats and Republicans? But the problem mentions the Independent as a member. Wait, the problem says "a Senate committee has 5 Democrats, 5 Republicans, and 1 Independent". So maybe the parties are Democrats, Republicans, and the Independent is not a party. Therefore, the condition is that all Democrats sit together, all Republicans sit together, and the Independent can be anywhere. But the problem says "all the members of each party all sit next to each other". So if the Independent is not a party, then they don't have to sit together. But the problem says "each party", so maybe the Independent is considered a party? Or maybe the problem is that the Independent is a party? Hmm.

Wait, the problem says "all the members of each party all sit next to each other". So if there are three parties: Democrats, Republicans, and Independent. Then each party's members must sit together. Therefore, the Independent, being a party of one, must sit together. Which is trivial. Therefore, the answer is as I calculated before: 2 * 5! * 5!.

But maybe the problem is that the Independent is not a party, so only Democrats and Republicans need to sit together. Then the problem would be different. But the problem says "each party", so if there are three parties, then the Independent is a party. But maybe the problem is that the Independent is not a party. Wait, the problem is from a math competition or something? Maybe the standard way is that the Independent is not considered a party. But the problem says "each party", so maybe the parties are Democrats and Republicans, and the Independent is not a party. Therefore, the condition is that Democrats sit together, Republicans sit together, and the Independent can be anywhere. But then the problem would have said "each of the two parties" or something. But the problem says "each party", and there are three parties. Wait, the problem says "a Senate committee has 5 Democrats, 5 Republicans, and 1 Independent". So maybe the parties are Democrats, Republicans, and the Independent is not a party. Therefore, the problem is that Democrats must sit together, Republicans must sit together, and the Independent can be anywhere. But then the problem would have said "each of the two parties" or "the two parties". But the problem says "each party", so maybe the Independent is considered a party. This is a critical ambiguity.

Wait, let me check the exact wording: "all the members of each party all sit next to each other". The committee has 5 Democrats, 5 Republicans, and 1 Independent. So the parties are Democrats, Republicans, and the Independent? Or is the Independent not a party? In real life, the Independent is not a party, but in the context of the problem, maybe they consider the Independent as a party. But the problem says "each party", so if there are three parties, then the answer is as I calculated. If there are two parties, then the answer is different.

But since the problem mentions the Independent as a separate member, maybe they are not considered a party. Therefore, the problem is that Democrats must sit together, Republicans must sit together, and the Independent can be anywhere. But then the problem would have said "each of the two parties" or "the two parties". However, the problem says "each party", so maybe the Independent is considered a party. Therefore, the answer is 2 * 5! * 5!.

But to resolve this ambiguity, let me think again. If the Independent is not a party, then the problem is that Democrats and Republicans must sit together, and the Independent can be anywhere. So how would that affect the calculation?

In that case, we have two blocks: Democrats and Republicans, and the Independent. So the total number of people is 5 + 5 + 1 = 11. But the two blocks (Democrats and Republicans) must sit together, and the Independent is separate. So how to arrange them around the table?

Wait, if the Democrats and Republicans are each blocks, and the Independent is a single person. So we have three "units": D block, R block, and I. So same as before. Therefore, the number of circular arrangements is (3-1)! = 2, multiplied by internal arrangements: 5! * 5! * 1. So same answer. Therefore, regardless of whether the Independent is considered a party or not, the answer is the same. Because even if the Independent is not a party, they are treated as a single unit. Therefore, the answer is 2 * 5! * 5!.

Therefore, maybe the answer is indeed 2 * 5! * 5!.

But let me check again. Suppose the problem had said that only Democrats and Republicans need to sit together, and the Independent can be anywhere. Then, how would that affect the calculation?

In that case, we have two blocks: D and R, and the Independent. So arranging three units around the table: D, R, I. So same as before. So the number of circular arrangements is (3-1)! = 2. Multiply by internal arrangements: 5! * 5! * 1. So same answer. Therefore, regardless of whether the Independent is considered a party or not, the answer is same. Therefore, the answer is 2 * 5! * 5!.

Therefore, I think that is the answer.

But let me check with another approach. Suppose we first consider the Democrats as a block, Republicans as a block, and the Independent as a single person. So total of 3 units. The number of circular arrangements is (3-1)! = 2. Then, for each arrangement, we can permute the Democrats in 5! ways, Republicans in 5! ways, and the Independent in 1 way. So total is 2 * 5! * 5!.

Alternatively, if we think of the Independent as part of the Democrats or Republicans? No, because they are separate. So that's not the case.

Alternatively, suppose we first arrange the Democrats and Republicans as blocks, and then place the Independent. But since it's a circular table, arranging the blocks and the Independent.

But again, same as before.

Alternatively, think of the entire seating. Since the Democrats are together, they form a block. Similarly, the Republicans form a block. The Independent is alone. So the seating is a circular arrangement of these three blocks. So the number of ways is (number of circular permutations of the blocks) * (internal permutations). Which is 2 * 5! * 5!.

Therefore, I think this is correct.

But let me check with numbers. Suppose we have smaller numbers. Let's say 1 Democrat, 1 Republican, and 1 Independent. How many ways?

If all must sit together, but since each party is a single person. So the number of circular arrangements. Since there are three blocks (each of size 1). The number of circular arrangements is (3-1)! = 2. Multiply by internal arrangements: 1! * 1! * 1! = 1. So total 2. Which is correct. Because in a circular table with three people, there are (3-1)! = 2 arrangements. So that works.

Another example: 2 Democrats, 1 Republican, 1 Independent. So blocks: D (2), R (1), I (1). Number of circular arrangements: (3-1)! = 2. Multiply by internal arrangements: 2! * 1! * 1! = 2. Total 2 * 2 = 4. Let's check. Fix the D block. Then the other two blocks (R and I) can be arranged in 2! ways. So 2 ways. For each, the D block can be arranged in 2! ways. So total 2 * 2 = 4. Which is correct. For example, if the D block is AB, then R and I can be arranged as R-I or I-R. So two arrangements. And AB can be BA, so total 2*2=4. Which is correct. So that seems to check out.

Therefore, returning to the original problem, the answer should be 2 * 5! * 5!.

But let me check if there's a different interpretation where the Independent is not considered a block. For example, if the problem requires that Democrats sit together, Republicans sit together, and the Independent is not required to sit with anyone. Then, how would that affect the calculation?

In that case, we have two blocks: D and R, and the Independent. So arranging three units around the table: D, R, I. Which is same as before. So same answer. Therefore, regardless of whether the Independent is considered a party or not, the answer is same. Therefore, the answer is 2 * 5! * 5!.

Therefore, I think that is the answer.

But let me check once more. Suppose the problem had said that only Democrats and Republicans need to sit together, and the Independent can be anywhere. Then, the number of ways would be:

First, treat D and R as blocks. So two blocks. Then, the Independent is a single person. So arranging three units: D, R, I. Which is same as before. So (3-1)! = 2. Multiply by internal arrangements: 5! * 5! * 1. So same answer. Therefore, regardless of the Independent's status, the answer is same. Therefore, the answer is 2 * 5! * 5!.

Therefore, I think that is the answer. So the final answer is 2 * 5! * 5!.

Calculating that, 5! is 120, so 120 * 120 = 14,400. Multiply by 2: 28,800. So 28,800 ways.

But let me check if there's a different approach where the answer is different. For example, suppose we first consider the seating of the Democrats and Republicans as blocks, and then place the Independent.

But since the table is circular, we can fix the position of one block to eliminate rotational symmetry. Let's fix the Democrats block. Then, the Republicans block and the Independent can be arranged in 2! ways around the table. Then, the Democrats can be arranged in 5! ways, the Republicans in 5! ways, and the Independent in 1 way. So total is 2! * 5! * 5! = 2 * 120 * 120 = 28,800. Same result.

Alternatively, if we didn't fix the Democrats block, then the number of ways to arrange the blocks is (number of circular permutations) = (3-1)! = 2. Then same result.

Therefore, I think that is correct.

Therefore, the answer is 2 * 5! * 5! = 28,800. So boxed answer is \boxed{28800}.

**Final Answer**
\boxed{28800}

To solve the problem of determining how many ways the Senate committee can sit around a circular table with the condition that all members of each party sit next to each other, we proceed step by step.

---

### **Understanding the Problem**

We are given:

- **5 Democrats**
- **5 Republicans**
- **1 Independent**

The condition is that **all members of each party must sit next to each other**. This means:

- The **5 Democrats** must form a **block**.
- The **5 Republicans** must form another **block**.
- The **1 Independent** is a **single person**, so they form a **block of size 1**.

Thus, we are arranging **three blocks** around a **circular table**.

---

### **Step 1: Circular Permutations of Blocks**

In circular arrangements, the number of distinct ways to arrange **k** distinguishable blocks is:

$$
(k - 1)!
$$

Here, we have **3 blocks** (D, R, I), so the number of circular arrangements is:

$$
(3 - 1)! = 2
$$

---

### **Step 2: Internal Arrangements Within Each Block**

Each block can be internally permuted:

- **Democrats block**: 5 people → $5!$ ways
- **Republicans block**: 5 people → $5!$ ways
- **Independent block**: 1 person → $1!$ way

So, the total number of internal arrangements is:

$$
5! \times 5! \times 1! = 120 \times 120 \times 1 = 14,400
$$

---

### **Step 3: Total Number of Arrangements**

Now, multiply the number of circular arrangements of the blocks by the number of internal arrangements:

$$
\text{Total} = 2 \times 5! \times 5! = 2 \times 120 \times 120 = 28,800
$$

---

### **Final Answer**

$$
\boxed{28800}
$$

Solution: Choose any seat in which to place the Independent -- it doesn't matter which seat that we choose, since we can rotate the table. Once the Independent's seat has been chosen, either all the Democrats sit to their left and all the Republicans sit to their right, or the other way around. Either way, there are $5!$ ways to put the Democrats in their seats, and $5!$ ways to put the Republicans in their seats. So, the total number of ways to seat the people around the table is $2\cdot5!\cdot5!=2\cdot120\cdot120=\boxed{28800}$.


Explanation:
Okay, so I need to find $ z^{85} + \frac{1}{z^{85}} $ given that $ z + \frac{1}{z} = \frac{1 + \sqrt{5}}{2} $. Hmm, let me think about how to approach this.

First, I remember that expressions like $ z + \frac{1}{z} $ often relate to trigonometric identities or maybe roots of unity. But since the given value is $ \frac{1 + \sqrt{5}}{2} $, which is the golden ratio, maybe there's a connection there. Alternatively, maybe I can use recurrence relations or some kind of recursive formula to compute higher powers?

Let me recall that if $ z + \frac{1}{z} = a $, then $ z^n + \frac{1}{z^n} $ can be found using recurrence relations. For example, if I let $ a_n = z^n + \frac{1}{z^n} $, then there's a recurrence relation that connects $ a_n $ with previous terms. Specifically, I think the relation is $ a_n = a_1 \cdot a_{n-1} - a_{n-2} $. Let me check that.

Let me verify with small n. Let's take n=2. Then $ a_2 = z^2 + 1/z^2 $. On the other hand, $ a_1 \cdot a_1 = (z + 1/z)^2 = z^2 + 2 + 1/z^2 $. Therefore, $ a_2 = a_1^2 - 2 $. So, if I use the recurrence $ a_n = a_1 \cdot a_{n-1} - a_{n-2} $, then for n=2, it would be $ a_2 = a_1 \cdot a_1 - a_0 $. But what is $ a_0 $? $ z^0 + 1/z^0 = 1 + 1 = 2 $. So, yes, $ a_2 = a_1^2 - 2 $, which matches. Similarly, for n=3, $ a_3 = a_1 \cdot a_2 - a_1 $. Let me check that.

Alternatively, $ a_3 = z^3 + 1/z^3 $. Let me compute $ (z + 1/z)^3 = z^3 + 3z + 3/z + 1/z^3 = a_3 + 3a_1 $. Therefore, $ a_3 = (a_1)^3 - 3a_1 $. But according to the recurrence $ a_3 = a_1 \cdot a_2 - a_1 $. Since $ a_2 = a_1^2 - 2 $, then $ a_1 \cdot a_2 - a_1 = a_1(a_1^2 - 2) - a_1 = a_1^3 - 2a_1 - a_1 = a_1^3 - 3a_1 $, which matches. So yes, the recurrence relation is correct. Therefore, the general recurrence is $ a_n = a_1 \cdot a_{n-1} - a_{n-2} $, with $ a_0 = 2 $, $ a_1 = \frac{1 + \sqrt{5}}{2} $.

Therefore, if I can compute $ a_{85} $, that would be the answer. However, computing $ a_{85} $ directly using this recurrence would be tedious. Maybe there's a pattern or periodicity? Or perhaps we can find a closed-form expression?

Alternatively, since $ z + 1/z = \frac{1 + \sqrt{5}}{2} $, maybe we can find z in terms of complex numbers? Let me try solving for z.

Let me set $ z + \frac{1}{z} = \frac{1 + \sqrt{5}}{2} $. Multiply both sides by z: $ z^2 + 1 = \frac{1 + \sqrt{5}}{2} z $. Then rearranged: $ z^2 - \frac{1 + \sqrt{5}}{2} z + 1 = 0 $. Let me compute the discriminant: $ D = \left( \frac{1 + \sqrt{5}}{2} \right)^2 - 4 \times 1 \times 1 $.

Calculating $ D $:

First, square $ \frac{1 + \sqrt{5}}{2} $:

$ \left( \frac{1 + \sqrt{5}}{2} \right)^2 = \frac{1 + 2\sqrt{5} + 5}{4} = \frac{6 + 2\sqrt{5}}{4} = \frac{3 + \sqrt{5}}{2} $.

Then subtract 4: $ D = \frac{3 + \sqrt{5}}{2} - 4 = \frac{3 + \sqrt{5} - 8}{2} = \frac{-5 + \sqrt{5}}{2} $.

Hmm, so the discriminant is $ \frac{-5 + \sqrt{5}}{2} $. Let me check if that's negative. Since $ \sqrt{5} \approx 2.236, so -5 + 2.236 ≈ -2.764, which is negative. Therefore, z is complex. Therefore, z is a complex number on the unit circle? Because if z is on the unit circle, then $ |z| = 1 $, so $ 1/z = \overline{z} $, and $ z + 1/z = 2 \Re(z) $. Therefore, if $ z + 1/z = \frac{1 + \sqrt{5}}{2} $, then $ 2 \Re(z) = \frac{1 + \sqrt{5}}{2} $, so $ \Re(z) = \frac{1 + \sqrt{5}}{4} \approx \frac{1 + 2.236}{4} ≈ 0.809 $, which is between -1 and 1, so that's okay. Therefore, z is on the unit circle. Therefore, we can write z as $ e^{i\theta} $, so that $ z + 1/z = 2 \cos \theta $. Therefore, $ 2 \cos \theta = \frac{1 + \sqrt{5}}{2} $, so $ \cos \theta = \frac{1 + \sqrt{5}}{4} $. Wait, but $ \frac{1 + \sqrt{5}}{4} $ is approximately 0.809/2? Wait, no, $ \frac{1 + \sqrt{5}}{2} $ is approximately 1.618/2 ≈ 0.809. Wait, no, $ \frac{1 + \sqrt{5}}{2} $ is the golden ratio φ ≈ 1.618, so $ 2 \cos \theta = φ $, so $ \cos \theta = φ / 2 ≈ 0.809 $, which is correct. Therefore, θ is an angle such that $ \cos \theta = \frac{1 + \sqrt{5}}{4} \times 2? Wait, no:

Wait, $ z + 1/z = 2 \cos \theta $, so $ 2 \cos \theta = \frac{1 + \sqrt{5}}{2} $, so $ \cos \theta = \frac{1 + \sqrt{5}}{4} $. Wait, but $ \frac{1 + \sqrt{5}}{4} $ is approximately (1 + 2.236)/4 ≈ 0.809, which is correct. So θ is arccos( (1 + √5)/4 ).

But maybe there's a better way to handle this. Since z is on the unit circle, then $ z^n + 1/z^n = 2 \cos(n\theta) $. Therefore, $ z^{85} + 1/z^{85} = 2 \cos(85\theta) $. Therefore, if I can find θ, then compute 85θ, and take cosine. However, θ is arccos( (1 + √5)/4 ). But maybe there's a relation between θ and some known angle? Or perhaps we can find θ in terms of π?

Alternatively, maybe we can find a recurrence relation for $ a_n = 2 \cos(n\theta) $, which is similar to the previous recurrence. Since $ a_n = 2 \cos(n\theta) $, and we know that $ a_1 = 2 \cos \theta = \frac{1 + \sqrt{5}}{2} $. Therefore, the recurrence relation $ a_n = a_1 a_{n-1} - a_{n-2} $ is satisfied by $ a_n = 2 \cos(n\theta) $. Therefore, if we can compute $ a_{85} $, that would be the answer. However, computing $ a_{85} $ using the recurrence might be tedious, but perhaps there's a pattern or periodicity?

Alternatively, since the recurrence is linear, maybe we can find a closed-form solution? Let me think. The recurrence is similar to the Fibonacci sequence? Let me check.

Let me denote $ a_n = z^n + 1/z^n $, with $ a_1 = \phi $, where $ \phi = \frac{1 + \sqrt{5}}{2} $. The recurrence is $ a_n = \phi a_{n-1} - a_{n-2} $. Let me check if this recurrence has a characteristic equation. Let me suppose that the solution is of the form $ r^n $. Then substituting into the recurrence:

$ r^n = \phi r^{n-1} - r^{n-2} $

Divide both sides by $ r^{n-2} $:

$ r^2 = \phi r - 1 $

Therefore, the characteristic equation is $ r^2 - \phi r + 1 = 0 $. Let me compute the roots:

Discriminant D = $ \phi^2 - 4 $. Since $ \phi = \frac{1 + \sqrt{5}}{2} $, then $ \phi^2 = \left( \frac{1 + \sqrt{5}}{2} \right)^2 = \frac{1 + 2\sqrt{5} + 5}{4} = \frac{6 + 2\sqrt{5}}{4} = \frac{3 + \sqrt{5}}{2} $. Therefore, D = $ \frac{3 + \sqrt{5}}{2} - 4 = \frac{3 + \sqrt{5} - 8}{2} = \frac{-5 + \sqrt{5}}{2} $, which is the same discriminant as before. Therefore, the roots are:

$ r = \frac{ \phi \pm \sqrt{D} }{2} $. Wait, but since D is negative, the roots are complex. Let me compute them:

Let me denote $ D = \frac{-5 + \sqrt{5}}{2} $. Let me compute sqrt(D). Wait, but since D is negative, sqrt(D) is imaginary. Let me write D as $ - \frac{5 - \sqrt{5}}{2} $. Therefore, sqrt(D) = i sqrt( (5 - sqrt(5))/2 ). Therefore, the roots are:

$ r = \frac{ \phi \pm i \sqrt{ \frac{5 - \sqrt{5}}{2} } }{2} $.

But this seems complicated. However, since we know that the solutions to the recurrence are $ z^n + 1/z^n $, which are real numbers, and since the characteristic equation has complex roots, which are conjugates, so the general solution is $ a_n = A r_1^n + B r_2^n $, but since $ r_1 $ and $ r_2 $ are complex conjugates, and since the original sequence $ a_n $ is real, this would result in a real sequence. However, since we already know that $ a_n = 2 \cos(n\theta) $, which is also real, so maybe this approach isn't helpful for computing $ a_{85} $.

Alternatively, maybe we can find a relation between $ a_n $ and Fibonacci numbers? Because the recurrence is similar to the Fibonacci recurrence, but with coefficients. Let me check.

The Fibonacci sequence has the recurrence $ F_n = F_{n-1} + F_{n-2} $, but here we have $ a_n = \phi a_{n-1} - a_{n-2} $. So it's a different recurrence. However, maybe we can find a relation between $ a_n $ and Fibonacci numbers?

Alternatively, since $ \phi $ is the golden ratio, which satisfies $ \phi^2 = \phi + 1 $, maybe we can use that property?

Let me note that $ \phi = \frac{1 + \sqrt{5}}{2} $, so $ \phi^2 = \frac{1 + 2\sqrt{5} + 5}{4} = \frac{6 + 2\sqrt{5}}{4} = \frac{3 + \sqrt{5}}{2} $. Also, $ \phi^2 = \phi + 1 $, which is a known identity. Let me verify:

$ \phi + 1 = \frac{1 + \sqrt{5}}{2} + 1 = \frac{3 + \sqrt{5}}{2} $, which matches $ \phi^2 $. So yes, $ \phi^2 = \phi + 1 $.

Therefore, maybe we can use this to find a pattern in the sequence $ a_n $.

Alternatively, since $ a_n = z^n + 1/z^n $, and z satisfies $ z + 1/z = \phi $, then z is a root of the equation $ x^2 - \phi x + 1 = 0 $, which we already saw. Therefore, z satisfies $ z^2 = \phi z - 1 $. Therefore, we can use this to find a recurrence for $ z^n $.

But since $ a_n = z^n + 1/z^n $, and $ z^n = \phi z^{n-1} - z^{n-2} $, similarly for $ 1/z^n = \phi / z^{n-1} - 1/z^{n-2} $. Therefore, adding them together:

$ a_n = \phi a_{n-1} - a_{n-2} $, which is the same recurrence as before. Therefore, we can use this recurrence to compute $ a_n $.

But computing up to n=85 would take a while. However, maybe there's a periodicity? Let me check for small n.

Let me compute the first few terms:

Given $ a_0 = 2 $, $ a_1 = \phi $.

Compute $ a_2 = \phi a_1 - a_0 = \phi^2 - 2 $. Since $ \phi^2 = \phi + 1 $, so $ a_2 = \phi + 1 - 2 = \phi - 1 $. But $ \phi - 1 = \frac{1 + \sqrt{5}}{2} - 1 = \frac{-1 + \sqrt{5}}{2} $.

Then $ a_3 = \phi a_2 - a_1 = \phi (\phi - 1) - \phi = \phi^2 - \phi - \phi = \phi^2 - 2\phi $. Since $ \phi^2 = \phi + 1 $, substitute: $ \phi + 1 - 2\phi = -\phi + 1 = 1 - \phi $.

Then $ a_4 = \phi a_3 - a_2 = \phi (1 - \phi) - (\phi - 1) = \phi - \phi^2 - \phi + 1 = -\phi^2 + 1 $. Again, $ \phi^2 = \phi + 1 $, so $ -(\phi + 1) + 1 = -\phi $.

Then $ a_5 = \phi a_4 - a_3 = \phi (-\phi) - (1 - \phi) = -\phi^2 - 1 + \phi = -(\phi + 1) - 1 + \phi = -\phi -1 -1 + \phi = -2 $.

Then $ a_6 = \phi a_5 - a_4 = \phi (-2) - (-\phi) = -2\phi + \phi = -\phi $.

Then $ a_7 = \phi a_6 - a_5 = \phi (-\phi) - (-2) = -\phi^2 + 2 = -(\phi + 1) + 2 = -\phi -1 + 2 = 1 - \phi $.

Wait, $ a_7 = 1 - \phi $, which is the same as $ a_3 $. Then $ a_8 = \phi a_7 - a_6 = \phi (1 - \phi) - (-\phi) = \phi - \phi^2 + \phi = 2\phi - \phi^2 $. Since $ \phi^2 = \phi + 1 $, substitute: $ 2\phi - \phi - 1 = \phi - 1 $, which is $ a_2 $. Then $ a_9 = \phi a_8 - a_7 = \phi (\phi - 1) - (1 - \phi) = \phi^2 - \phi -1 + \phi = \phi^2 - 1 = (\phi + 1) - 1 = \phi $. Which is $ a_1 $. Then $ a_{10} = \phi a_9 - a_8 = \phi^2 - (\phi - 1) = \phi + 1 - \phi + 1 = 2 $, which is $ a_0 $.

So, the sequence is periodic with period 10? Let me check:

a0 = 2

a1 = φ

a2 = φ - 1

a3 = 1 - φ

a4 = -φ

a5 = -2

a6 = -φ

a7 = 1 - φ

a8 = φ - 1

a9 = φ

a10 = 2

Yes! So the sequence repeats every 10 terms. Therefore, the period is 10. Therefore, $ a_n $ is periodic with period 10. Therefore, $ a_{n} = a_{n \mod 10} $, but we have to be careful with the modulus. Let me check:

For example, a10 = a0, a11 = a1, etc. Therefore, the period is 10. Therefore, to compute $ a_{85} $, we can compute 85 mod 10. 85 divided by 10 is 8*10=80, remainder 5. Therefore, 85 mod 10 is 5. Therefore, $ a_{85} = a_5 = -2 $. Therefore, the answer is -2?

Wait, but let me check again. Let me verify with the earlier terms:

a0 = 2

a1 = φ

a2 = φ - 1

a3 = 1 - φ

a4 = -φ

a5 = -2

a6 = -φ

a7 = 1 - φ

a8 = φ - 1

a9 = φ

a10 = 2

Yes, so the cycle is 10 terms. Therefore, 85 divided by 10 is 8*10=80, remainder 5. Therefore, a85 = a5 = -2. Therefore, the answer is -2? Wait, but let me check with another term. For example, a10 = 2, which is a0. Then a11 = a1, which would be φ. Let me compute a11 using the recurrence:

a11 = φ*a10 - a9 = φ*2 - φ = 2φ - φ = φ, which matches. Similarly, a12 = φ*a11 - a10 = φ^2 - 2 = φ + 1 - 2 = φ - 1, which is a2. So yes, the cycle is indeed 10. Therefore, 85 mod 10 is 5, so a85 = a5 = -2. Therefore, the answer is -2.

But let me check with another approach. Since z is on the unit circle, and $ z + 1/z = 2 \cos \theta = \phi $, so $ \cos \theta = \phi / 2 \approx 0.809 $. Then, $ z^{85} + 1/z^{85} = 2 \cos(85\theta) $. If the sequence is periodic with period 10, then 85θ is equivalent to 5θ modulo 2π? Wait, but how does that relate?

Alternatively, since the sequence a_n is periodic with period 10, then 85θ is equivalent to 5θ modulo 2π? Wait, but how does that work? Let me think. If a_n = 2 cos(nθ), and a_n is periodic with period 10, then 2 cos(nθ) = 2 cos((n + 10)θ). Therefore, cos(nθ) = cos((n + 10)θ). Therefore, this implies that 10θ is a multiple of 2π, or θ is a multiple of 2π/10 = π/5. Wait, but θ is arccos(φ/2). Let me check if 10θ is a multiple of 2π?

Wait, if θ is such that 10θ = 2π k for some integer k, then θ = π k / 5. Let me check if θ = π/5? Let me compute cos(π/5). cos(π/5) is (1 + √5)/4 * 2? Wait, cos(π/5) is equal to (1 + √5)/4 * 2? Wait, cos(π/5) is known to be (sqrt(5) + 1)/4 * 2? Let me recall exact value.

Yes, cos(π/5) = (1 + √5)/4 * 2? Wait, cos(π/5) is equal to (sqrt(5) + 1)/4 multiplied by 2? Let me compute cos(π/5):

cos(π/5) = cos(36°) ≈ 0.8090, which is exactly (1 + √5)/4 * 2? Let me compute (1 + √5)/4 * 2 = (1 + √5)/2 ≈ (1 + 2.236)/2 ≈ 1.618/2 ≈ 0.809, which matches. Therefore, cos(π/5) = φ/2. Therefore, θ = π/5. Therefore, θ = π/5. Therefore, 10θ = 2π. Therefore, cos(10θ) = cos(2π) = 1. Therefore, indeed, the sequence a_n is periodic with period 10. Therefore, a_{n + 10} = a_n. Therefore, a_{85} = a_{5} = -2. Therefore, the answer is -2.

Therefore, the answer is \boxed{-2}.

But let me check once again with θ = π/5. Then, z = e^{iπ/5} or e^{-iπ/5}. Therefore, z^85 = e^{i85π/5} = e^{i17π} = e^{i(16π + π)} = e^{iπ} = -1. Similarly, 1/z^85 = e^{-i17π} = -1. Therefore, z^85 + 1/z^85 = -1 + (-1) = -2. Which matches. Therefore, this confirms that the answer is indeed -2.

Therefore, the answer is \boxed{-2}.

**Final Answer**
\boxed{-2}

To solve the problem, we are given the equation:

$$
z + \frac{1}{z} = \frac{1 + \sqrt{5}}{2}
$$

and asked to find:

$$
z^{85} + \frac{1}{z^{85}}
$$

---

### Step 1: Recognize the Structure

Let us define:

$$
a_n = z^n + \frac{1}{z^n}
$$

We are given $ a_1 = \frac{1 + \sqrt{5}}{2} $, which is the **golden ratio**, denoted $ \phi $. This value is known to satisfy the identity:

$$
\phi^2 = \phi + 1
$$

This identity will be useful later.

---

### Step 2: Derive a Recurrence Relation

We can derive a recurrence relation for $ a_n $ using the identity:

$$
a_n = a_1 a_{n-1} - a_{n-2}
$$

This is a second-order linear recurrence relation with constant coefficients. The initial conditions are:

- $ a_0 = z^0 + \frac{1}{z^0} = 1 + 1 = 2 $
- $ a_1 = \phi $

---

### Step 3: Compute the First Few Terms

Using the recurrence:

- $ a_0 = 2 $
- $ a_1 = \phi $
- $ a_2 = \phi a_1 - a_0 = \phi^2 - 2 = (\phi + 1) - 2 = \phi - 1 $
- $ a_3 = \phi a_2 - a_1 = \phi(\phi - 1) - \phi = \phi^2 - \phi - \phi = \phi^2 - 2\phi = (\phi + 1) - 2\phi = 1 - \phi $
- $ a_4 = \phi a_3 - a_2 = \phi(1 - \phi) - (\phi - 1) = \phi - \phi^2 - \phi + 1 = -\phi^2 + 1 = -(\phi + 1) + 1 = -\phi $
- $ a_5 = \phi a_4 - a_3 = \phi(-\phi) - (1 - \phi) = -\phi^2 - 1 + \phi = -(\phi + 1) - 1 + \phi = -2 $

Continuing this process, we find that the sequence becomes periodic with period 10:

$$
\begin{align*}
a_0 &= 2 \\
a_1 &= \phi \\
a_2 &= \phi - 1 \\
a_3 &= 1 - \phi \\
a_4 &= -\phi \\
a_5 &= -2 \\
a_6 &= -\phi \\
a_7 &= 1 - \phi \\
a_8 &= \phi - 1 \\
a_9 &= \phi \\
a_{10} &= 2 \\
\end{align*}
$$

So the sequence is periodic with period 10.

---

### Step 4: Use the Periodicity to Find $ a_{85} $

Since the sequence is periodic with period 10, we compute:

$$
85 \mod 10 = 5
$$

Thus,

$$
a_{85} = a_5 = -2
$$

---

### Step 5: Final Answer

$$
z^{85} + \frac{1}{z^{85}} = a_{85} = \boxed{-2}
$$

[Assistant]
Explanation:
Okay, so I need to find $ z^{85} + \frac{1}{z^{85}} $ given that $ z + \frac{1}{z} = \frac{1 + \sqrt{5}}{2} $. Hmm, let me think about how to approach this.

First, I remember that expressions like $ z + \frac{1}{z} $ often relate to trigonometric identities or maybe roots of unity. But since the given value is $ \frac{1 + \sqrt{5}}{2} $, which is the golden ratio, maybe there's a connection there. Alternatively, maybe I can use recurrence relations or some kind of recursive formula to compute higher powers?

Let me recall that if $ z + \frac{1}{z} = a $, then $ z^n + \frac{1}{z^n} $ can be found using recurrence relations. For example, if I let $ a_n = z^n + \frac{1}{z^n} $, then there's a recurrence relation that connects $ a_n $ with previous terms. Specifically, I think the relation is $ a_n = a_1 \cdot a_{n-1} - a_{n-2} $. Let me check that.

Let me verify with small n. Let's take n=2. Then $ a_2 = z^2 + 1/z^2 $. On the other hand, $ a_1 \cdot a_1 = (z + 1/z)^2 = z^2 + 2 + 1/z^2 $. Therefore, $ a_2 = a_1^2 - 2 $. So, if I use the recurrence $ a_n = a_1 \cdot a_{n-1} - a_{n-2} $, then for n=2, it would be $ a_2 = a_1 \cdot a_1 - a_0 $. But what is $ a_0 $? $ z^0 + 1/z^0 = 1 + 1 = 2 $. So, yes, $ a_2 = a_1^2 - 2 $, which matches. Similarly, for n=3, $ a_3 = a_1 \cdot a_2 - a_1 $. Let me check that.

Alternatively, $ a_3 = z^3 + 1/z^3 $. Let me compute $ (z + 1/z)^3 = z^3 + 3z + 3/z + 1/z^3 = a_3 + 3a_1 $. Therefore, $ a_3 = (a_1)^3 - 3a_1 $. But according to the recurrence $ a_3 = a_1 \cdot a_2 - a_1 $. Since $ a_2 = a_1^2 - 2 $, then $ a_1 \cdot a_2 - a_1 = a_1(a_1^2 - 2) - a_1 = a_1^3 - 2a_1 - a_1 = a_1^3 - 3a_1 $, which matches. So yes, the recurrence relation is correct. Therefore, the general recurrence is $ a_n = a_1 \cdot a_{n-1} - a_{n-2} $, with $ a_0 = 2 $, $ a_1 = \frac{1 + \sqrt{5}}{2} $.

Therefore, if I can compute $ a_{85} $, that would be the answer. However, computing $ a_{85} $ directly using this recurrence would be tedious. Maybe there's a pattern or periodicity? Or perhaps we can find a closed-form expression?

Alternatively, since $ z + 1/z = \frac{1 + \sqrt{5}}{2} $, maybe we can find z in terms of complex numbers? Let me try solving for z.

Let me set $ z + \frac{1}{z} = \frac{1 + \sqrt{5}}{2} $. Multiply both sides by z: $ z^2 + 1 = \frac{1 + \sqrt{5}}{2} z $. Then rearranged: $ z^2 - \frac{1 + \sqrt{5}}{2} z + 1 = 0 $. Let me compute the discriminant: $ D = \left( \frac{1 + \sqrt{5}}{2} \right)^2 - 4 \times 1 \times 1 $.

Calculating $ D $:

First, square $ \frac{1 + \sqrt{5}}{2} $:

$ \left( \frac{1 + \sqrt{5}}{2} \right)^2 = \frac{1 + 2\sqrt{5} + 5}{4} = \frac{6 + 2\sqrt{5}}{4} = \frac{3 + \sqrt{5}}{2} $.

Then subtract 4: $ D = \frac{3 + \sqrt{5}}{2} - 4 = \frac{3 + \sqrt{5} - 8}{2} = \frac{-5 + \sqrt{5}}{2} $.

Hmm, so the discriminant is $ \frac{-5 + \sqrt{5}}{2} $. Let me check if that's negative. Since $ \sqrt{5} \approx 2.236, so -5 + 2.236 ≈ -2.764, which is negative. Therefore, z is complex. Therefore, z is a complex number on the unit circle? Because if z is on the unit circle, then $ |z| = 1 $, so $ 1/z = \overline{z} $, and $ z + 1/z = 2 \Re(z) $. Therefore, if $ z + 1/z = \frac{1 + \sqrt{5}}{2} $, then $ 2 \Re(z) = \frac{1 + \sqrt{5}}{2} $, so $ \Re(z) = \frac{1 + \sqrt{5}}{4} \approx \frac{1 + 2.236}{4} ≈ 0.809 $, which is between -1 and 1, so that's okay. Therefore, z is on the unit circle. Therefore, we can write z as $ e^{i\theta} $, so that $ z + 1/z = 2 \cos \theta $. Therefore, $ 2 \cos \theta = \frac{1 + \sqrt{5}}{2} $, so $ \cos \theta = \frac{1 + \sqrt{5}}{4} $. Wait, but $ \frac{1 + \sqrt{5}}{4} $ is approximately 0.809/2? Wait, no, $ \frac{1 + \sqrt{5}}{2} $ is approximately 1.618/2 ≈ 0.809. Wait, no, $ \frac{1 + \sqrt{5}}{2} $ is the golden ratio φ ≈ 1.618, so $ 2 \cos \theta = φ $, so $ \cos \theta = φ / 2 ≈ 0.809 $, which is correct. Therefore, θ is an angle such that $ \cos \theta = \frac{1 + \sqrt{5}}{4} \times 2? Wait, no:

Wait, $ z + 1/z = 2 \cos \theta $, so $ 2 \cos \theta = \frac{1 + \sqrt{5}}{2} $, so $ \cos \theta = \frac{1 + \sqrt{5}}{4} $. Wait, but $ \frac{1 + \sqrt{5}}{4} $ is approximately (1 + 2.236)/4 ≈ 0.809, which is correct. So θ is arccos( (1 + √5)/4 ).

But maybe there's a better way to handle this. Since z is on the unit circle, then $ z^n + 1/z^n = 2 \cos(n\theta) $. Therefore, $ z^{85} + 1/z^{85} = 2 \cos(85\theta) $. Therefore, if I can find θ, then compute 85θ, and take cosine. However, θ is arccos( (1 + √5)/4 ). But maybe there's a relation between θ and some known angle? Or perhaps we can find θ in terms of π?

Alternatively, maybe we can find a recurrence relation for $ a_n = 2 \cos(n\theta) $, which is similar to the previous recurrence. Since $ a_n = 2 \cos(n\theta) $, and we know that $ a_1 = 2 \cos \theta = \frac{1 + \sqrt{5}}{2} $. Therefore, the recurrence relation $ a_n = a_1 a_{n-1} - a_{n-2} $ is satisfied by $ a_n = 2 \cos(n\theta) $. Therefore, if we can compute $ a_{85} $, that would be the answer. However, computing $ a_{85} $ using the recurrence might be tedious, but perhaps there's a pattern or periodicity?

Alternatively, since the recurrence is linear, maybe we can find a closed-form solution? Let me think. The recurrence is similar to the Fibonacci sequence? Let me check.

Let me denote $ a_n = z^n + 1/z^n $, with $ a_1 = \phi $, where $ \phi = \frac{1 + \sqrt{5}}{2} $. The recurrence is $ a_n = \phi a_{n-1} - a_{n-2} $. Let me check if this recurrence has a characteristic equation. Let me suppose that the solution is of the form $ r^n $. Then substituting into the recurrence:

$ r^n = \phi r^{n-1} - r^{n-2} $

Divide both sides by $ r^{n-2} $:

$ r^2 = \phi r - 1 $

Therefore, the characteristic equation is $ r^2 - \phi r + 1 = 0 $. Let me compute the roots:

Discriminant D = $ \phi^2 - 4 $. Since $ \phi = \frac{1 + \sqrt{5}}{2} $, then $ \phi^2 = \left( \frac{1 + \sqrt{5}}{2} \right)^2 = \frac{1 + 2\sqrt{5} + 5}{4} = \frac{6 + 2\sqrt{5}}{4} = \frac{3 + \sqrt{5}}{2} $. Therefore, D = $ \frac{3 + \sqrt{5}}{2} - 4 = \frac{3 + \sqrt{5} - 8}{2} = \frac{-5 + \sqrt{5}}{2} $, which is the same discriminant as before. Therefore, the roots are:

$ r = \frac{ \phi \pm \sqrt{D} }{2} $. Wait, but since D is negative, the roots are complex. Let me compute them:

Let me denote $ D = \frac{-5 + \sqrt{5}}{2} $. Let me compute sqrt(D). Wait, but since D is negative, sqrt(D) is imaginary. Let me write D as $ - \frac{5 - \sqrt{5}}{2} $. Therefore, sqrt(D) = i sqrt( (5 - sqrt(5))/2 ). Therefore, the roots are:

$ r = \frac{ \phi \pm i \sqrt{ \frac{5 - \sqrt{5}}{2} } }{2} $.

But this seems complicated. However, since we know that the solutions to the recurrence are $ z^n + 1/z^n $, which are real numbers, and since the characteristic equation has complex roots, which are conjugates, so the general solution is $ a_n = A r_1^n + B r_2^n $, but since $ r_1 $ and $ r_2 $ are complex conjugates, and since the original sequence $ a_n $ is real, this would result in a real sequence. However, since we already know that $ a_n = 2 \cos(n\theta) $, which is also real, so maybe this approach isn't helpful for computing $ a_{85} $.

Alternatively, maybe we can find a relation between $ a_n $ and Fibonacci numbers? Because the recurrence is similar to the Fibonacci recurrence, but with coefficients. Let me check.

The Fibonacci sequence has the recurrence $ F_n = F_{n-1} + F_{n-2} $, but here we have $ a_n = \phi a_{n-1} - a_{n-2} $. So it's a different recurrence. However, maybe we can find a relation between $ a_n $ and Fibonacci numbers?

Alternatively, since $ \phi $ is the golden ratio, which satisfies $ \phi^2 = \phi + 1 $, maybe we can use that property?

Let me note that $ \phi = \frac{1 + \sqrt{5}}{2} $, so $ \phi^2 = \frac{1 + 2\sqrt{5} + 5}{4} = \frac{6 + 2\sqrt{5}}{4} = \frac{3 + \sqrt{5}}{2} $. Also, $ \phi^2 = \phi + 1 $, which is a known identity. Let me verify:

$ \phi + 1 = \frac{1 + \sqrt{5}}{2} + 1 = \frac{3 + \sqrt{5}}{2} $, which matches $ \phi^2 $. So yes, $ \phi^2 = \phi + 1 $.

Therefore, maybe we can use this to find a pattern in the sequence $ a_n $.

Alternatively, since $ a_n = z^n + 1/z^n $, and z satisfies $ z + 1/z = \phi $, then z is a root of the equation $ x^2 - \phi x + 1 = 0 $, which we already saw. Therefore, z satisfies $ z^2 = \phi z - 1 $. Therefore, we can use this to find a recurrence for $ z^n $.

But since $ a_n = z^n + 1/z^n $, and $ z^n = \phi z^{n-1} - z^{n-2} $, similarly for $ 1/z^n = \phi / z^{n-1} - 1/z^{n-2} $. Therefore, adding them together:

$ a_n = \phi a_{n-1} - a_{n-2} $, which is the same recurrence as before. Therefore, we can use this recurrence to compute $ a_n $.

But computing up to n=85 would take a while. However, maybe there's a periodicity? Let me check for small n.

Let me compute the first few terms:

Given $ a_0 = 2 $, $ a_1 = \phi $.

Compute $ a_2 = \phi a_1 - a_0 = \phi^2 - 2 $. Since $ \phi^2 = \phi + 1 $, so $ a_2 = \phi + 1 - 2 = \phi - 1 $. But $ \phi - 1 = \frac{1 + \sqrt{5}}{2} - 1 = \frac{-1 + \sqrt{5}}{2} $.

Then $ a_3 = \phi a_2 - a_1 = \phi (\phi - 1) - \phi = \phi^2 - \phi - \phi = \phi^2 - 2\phi $. Since $ \phi^2 = \phi + 1 $, substitute: $ \phi + 1 - 2\phi = -\phi + 1 = 1 - \phi $.

Then $ a_4 = \phi a_3 - a_2 = \phi (1 - \phi) - (\phi - 1) = \phi - \phi^2 - \phi + 1 = -\phi^2 + 1 $. Again, $ \phi^2 = \phi + 1 $, so $ -(\phi + 1) + 1 = -\phi $.

Then $ a_5 = \phi a_4 - a_3 = \phi (-\phi) - (1 - \phi) = -\phi^2 - 1 + \phi = -(\phi + 1) - 1 + \phi = -\phi -1 -1 + \phi = -2 $.

Then $ a_6 = \phi a_5 - a_4 = \phi (-2) - (-\phi) = -2\phi + \phi = -\phi $.

Then $ a_7 = \phi a_6 - a_5 = \phi (-\phi) - (-2) = -\phi^2 + 2 = -(\phi + 1) + 2 = -\phi -1 + 2 = 1 - \phi $.

Wait, $ a_7 = 1 - \phi $, which is the same as $ a_3 $. Then $ a_8 = \phi a_7 - a_6 = \phi (1 - \phi) - (-\phi) = \phi - \phi^2 + \phi = 2\phi - \phi^2 $. Since $ \phi^2 = \phi + 1 $, substitute: $ 2\phi - \phi - 1 = \phi - 1 $, which is $ a_2 $. Then $ a_9 = \phi a_8 - a_7 = \phi (\phi - 1) - (1 - \phi) = \phi^2 - \phi -1 + \phi = \phi^2 - 1 = (\phi + 1) - 1 = \phi $. Which is $ a_1 $. Then $ a_{10} = \phi a_9 - a_8 = \phi^2 - (\phi - 1) = \phi + 1 - \phi + 1 = 2 $, which is $ a_0 $.

So, the sequence is periodic with period 10? Let me check:

a0 = 2

a1 = φ

a2 = φ - 1

a3 = 1 - φ

a4 = -φ

a5 = -2

a6 = -φ

a7 = 1 - φ

a8 = φ - 1

a9 = φ

a10 = 2

Yes! So the sequence repeats every 10 terms. Therefore, the period is 10. Therefore, $ a_n $ is periodic with period 10. Therefore, $ a_{n} = a_{n \mod 10} $, but we have to be careful with the modulus. Let me check:

For example, a10 = a0, a11 = a1, etc. Therefore, the period is 10. Therefore, to compute $ a_{85} $, we can compute 85 mod 10. 85 divided by 10 is 8*10=80, remainder 5. Therefore, 85 mod 10 is 5. Therefore, $ a_{85} = a_5 = -2 $. Therefore, the answer is -2?

Wait, but let me check again. Let me verify with the earlier terms:

a0 = 2

a1 = φ

a2 = φ - 1

a3 = 1 - φ

a4 = -φ

a5 = -2

a6 = -φ

a7 = 1 - φ

a8 = φ - 1

a9 = φ

a10 = 2

Yes, so the cycle is 10 terms. Therefore, 85 divided by 10 is 8*10=80, remainder 5. Therefore, a85 = a5 = -2. Therefore, the answer is -2? Wait, but let me check with another term. For example, a10 = 2, which is a0. Then a11 = a1, which would be φ. Let me compute a11 using the recurrence:

a11 = φ*a10 - a9 = φ*2 - φ = 2φ - φ = φ, which matches. Similarly, a12 = φ*a11 - a10 = φ^2 - 2 = φ + 1 - 2 = φ - 1, which is a2. So yes, the cycle is indeed 10. Therefore, 85 mod 10 is 5, so a85 = a5 = -2. Therefore, the answer is -2.

But let me check with another approach. Since z is on the unit circle, and $ z + 1/z = 2 \cos \theta = \phi $, so $ \cos \theta = \phi / 2 \approx 0.809 $. Then, $ z^{85} + 1/z^{85} = 2 \cos(85\theta) $. If the sequence is periodic with period 10, then 85θ is equivalent to 5θ modulo 2π? Wait, but how does that relate?

Alternatively, since the sequence a_n is periodic with period 10, then 85θ is equivalent to 5θ modulo 2π? Wait, but how does that work? Let me think. If a_n = 2 cos(nθ), and a_n is periodic with period 10, then 2 cos(nθ) = 2 cos((n + 10)θ). Therefore, cos(nθ) = cos((n + 10)θ). Therefore, this implies that 10θ is a multiple of 2π, or θ is a multiple of 2π/10 = π/5. Wait, but θ is arccos(φ/2). Let me check if 10θ is a multiple of 2π?

Wait, if θ is such that 10θ = 2π k for some integer k, then θ = π k / 5. Let me check if θ = π/5? Let me compute cos(π/5). cos(π/5) is (1 + √5)/4 * 2? Wait, cos(π/5) is equal to (1 + √5)/4 * 2? Wait, cos(π/5) is known to be (sqrt(5) + 1)/4 * 2? Let me recall exact value.

Yes, cos(π/5) = (1 + √5)/4 * 2? Wait, cos(π/5) is equal to (sqrt(5) + 1)/4 multiplied by 2? Let me compute cos(π/5):

cos(π/5) = cos(36°) ≈ 0.8090, which is exactly (1 + √5)/4 * 2? Let me compute (1 + √5)/4 * 2 = (1 + √5)/2 ≈ (1 + 2.236)/2 ≈ 1.618/2 ≈ 0.809, which matches. Therefore, cos(π/5) = φ/2. Therefore, θ = π/5. Therefore, θ = π/5. Therefore, 10θ = 2π. Therefore, cos(10θ) = cos(2π) = 1. Therefore, indeed, the sequence a_n is periodic with period 10. Therefore, a_{n + 10} = a_n. Therefore, a_{85} = a_{5} = -2. Therefore, the answer is -2.

Therefore, the answer is \boxed{-2}.

But let me check once again with θ = π/5. Then, z = e^{iπ/5} or e^{-iπ/5}. Therefore, z^85 = e^{i85π/5} = e^{i17π} = e^{i(16π + π)} = e^{iπ} = -1. Similarly, 1/z^85 = e^{-i17π} = -1. Therefore, z^85 + 1/z^85 = -1 + (-1) = -2. Which matches. Therefore, this confirms that the answer is indeed -2.

Therefore, the answer is \boxed{-2}.

**Final Answer**
\boxed{-2}

To solve the problem, we are given the equation:

$$
z + \frac{1}{z} = \frac{1 + \sqrt{5}}{2}
$$

and asked to find:

$$
z^{85} + \frac{1}{z^{85}}
$$

---

### Step 1: Recognize the Structure

Let us define:

$$
a_n = z^n + \frac{1}{z^n}
$$

We are given $ a_1 = \frac{1 + \sqrt{5}}{2} $, which is the **golden ratio**, denoted $ \phi $. This value is known to satisfy the identity:

$$
\phi^2 = \phi + 1
$$

This identity will be useful later.

---

### Step 2: Derive a Recurrence Relation

We can derive a recurrence relation for $ a_n $ using the identity:

$$
a_n = a_1 a_{n-1} - a_{n-2}
$$

This is a second-order linear recurrence relation with constant coefficients. The initial conditions are:

- $ a_0 = z^0 + \frac{1}{z^0} = 1 + 1 = 2 $
- $ a_1 = \phi $

---

### Step 3: Compute the First Few Terms

Using the recurrence:

- $ a_0 = 2 $
- $ a_1 = \phi $
- $ a_2 = \phi a_1 - a_0 = \phi^2 - 2 = (\phi + 1) - 2 = \phi - 1 $
- $ a_3 = \phi a_2 - a_1 = \phi(\phi - 1) - \phi = \phi^2 - \phi - \phi = \phi^2 - 2\phi = (\phi + 1) - 2\phi = 1 - \phi $
- $ a_4 = \phi a_3 - a_2 = \phi(1 - \phi) - (\phi - 1) = \phi - \phi^2 - \phi + 1 = -\phi^2 + 1 = -(\phi + 1) + 1 = -\phi $
- $ a_5 = \phi a_4 - a_3 = \phi(-\phi) - (1 - \phi) = -\phi^2 - 1 + \phi = -(\phi + 1) - 1 + \phi = -2 $

Continuing this process, we find that the sequence becomes periodic with period 10:

$$
\begin{align*}
a_0 &= 2 \\
a_1 &= \phi \\
a_2 &= \phi - 1 \\
a_3 &= 1 - \phi \\
a_4 &= -\phi \\
a_5 &= -2 \\
a_6 &= -\phi \\
a_7 &= 1 - \phi \\
a_8 &= \phi - 1 \\
a_9 &= \phi \\
a_{10} &= 2 \\
\end{align*}
$$

So the sequence is periodic with period 10.

---

### Step 4: Use the Periodicity to Find $ a_{85} $

Since the sequence is periodic with period 10, we compute:

$$
85 \mod 10 = 5
$$

Thus,

$$
a_{85} = a_5 = -2
$$

---

### Step 5: Final Answer

$$
z^{85} + \frac{1}{z^{85}} = a_{85} = \boxed{-2}
$$

Solution:
From $z + \frac{1}{z} = \frac{1 + \sqrt{5}}{2},$
\[z + \frac{1}{z} - \frac{1}{2} = \frac{\sqrt{5}}{2}.\]Squaring both sides, we end up with
\[z^2 - z + \frac{9}{4} - \frac{1}{z} + \frac{1}{z^2} = \frac{5}{4}.\]Then
\[z^2 - z + 1 - \frac{1}{z} + \frac{1}{z^2} = 0.\]Hence, $z^4 - z^3 + z^2 - z + 1 = 0.$  Then
\[(z + 1)(z^4 - z^3 + z^2 - z + 1) = 0,\]which expands as $z^5 + 1 = 0.$  This gives us $z^5 = -1.$

Therefore,
\[z^{85} + \frac{1}{z^{85}} = (z^5)^{17} + \frac{1}{(z^5)^{17}} = (-1)^{17} + \frac{1}{(-1)^{17}} = \boxed{-2}.\]