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 about how to approach this.

First, circular permutations can be tricky because rotations are considered the same. Normally, for n people sitting 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 the Independent. However, the Independent is only one person. So does that mean the Independent can be considered as a block of size 1? Or does the problem require that all members of each party are seated together, so the Independent is a party of one? Let me check the problem statement again: "all the members of each party all sit next to each other." Since there is only one Independent, that person is already sitting alone, so technically, the condition is satisfied because the single member is trivially sitting next to themselves? Maybe.

So, if we think of the three parties as three blocks: D (Democrats), R (Republicans), and I (Independent). But since the Independent is just one person, their block is just one person. So, how many ways can we arrange these three blocks around the table?

But since it's a circular table, arranging blocks around a circular table. Normally, for circular arrangements of k blocks, it's (k-1)! ways. But here, the blocks have internal arrangements. So, for each block, we need to consider the number of ways to arrange the members within the block.

So, first, arrange the three blocks around the table. Since rotations are considered the same, it's (3-1)! = 2! = 2 ways. Then, for each block, multiply by the number of internal arrangements.

But wait, the Independent is a single person, so there's only 1 way to arrange them. The Democrats are 5 people, so they can be arranged in 5! ways. Similarly, the Republicans are 5 people, so 5! ways. So total number would be 2! * 5! * 5! * 1?

Wait, that seems too small. Let me check again. Wait, if we have three blocks: D, R, I. Since it's a circular table, arranging three blocks would be (3-1)! = 2 ways. Then multiply by the permutations within each block. So that would be 2 * 5! * 5! * 1. But is that correct?

But wait, let me think again. Suppose we fix the position of one block to account for the circular table. For example, fix the Democrats block in a certain position. Then arrange the other two blocks (Republicans and Independent) around the table. Since there are 2! ways to arrange the remaining two blocks. Then, within the Democrats block, 5! arrangements, Republicans block 5! arrangements, and the Independent is fixed. So total is 2! * 5! * 5!.

But why does fixing the Democrats block account for the circular rotation? Because in circular permutations, fixing one position removes rotational symmetry. So yes, that seems right. So the answer would be 2 * 5! * 5!.

Wait, but let me check if there's another way to think about it. Alternatively, sometimes when dealing with circular permutations with blocks, the formula is (number of blocks - 1)! multiplied by the product of the internal arrangements. So here, the number of blocks is 3, so (3-1)! * 5! * 5! * 1. So that's 2! * 5! * 5! = 2 * 120 * 120 = 2 * 14400 = 28800. Is that the answer? Hmm, but maybe I'm missing something?

Wait, let me consider the Independent. Since the Independent is a single person, but is there a different way they can be arranged? Or is the Independent considered as a block of size 1? Well, if we treat them as a block, then arranging the three blocks (D, R, I) around the table, which is (3-1)! = 2 ways. Then multiply by the internal arrangements. For D: 5!, R: 5!, I: 1! = 1. So total is 2 * 5! * 5! = 28800. So maybe that is correct?

But let me think again. What if the Independent is considered as a separate block, so the three blocks are D, R, and I. But in circular arrangements, if you have three blocks, how many ways are there to arrange them? It's (3-1)! = 2. But if the blocks are different, then that's correct. So yes, two ways to arrange the blocks.

But wait, suppose that the Independent is not a block? Wait, no. Because the problem says that all members of each party must sit next to each other. Since the Independent is a party by themselves, they have to sit next to themselves. Which is trivially true. So the Independent is a block of size 1.

But let me check if there's another interpretation. Suppose that the problem requires that each party (Democrats, Republicans, and the Independent) are all seated together. However, the Independent is only one person, so they are already seated alone. Therefore, the only way for the Independent to be seated together is if they are seated alone. Therefore, the problem is equivalent to arranging Democrats, Republicans, and Independent as blocks. So the answer should be (number of ways to arrange the blocks) multiplied by internal arrangements.

But let me check another way. Suppose we think of the entire seating as three blocks: D, R, I. Since it's a circular table, the number of ways to arrange these blocks is (3-1)! = 2. Then, for each block, the number of arrangements is 5! for Democrats, 5! for Republicans, and 1! for the Independent. So total is 2 * 5! * 5!.

But perhaps there is a mistake here. Let me think. What if we didn't fix the blocks? For example, if there are three blocks, the number of circular permutations is (number of blocks - 1)! So 2! = 2. So that seems correct. Then multiplied by internal permutations. So 2 * 5! * 5!.

But let me see if I can think of an example with smaller numbers. Let's say there are 1 Democrat, 1 Republican, and 1 Independent. So total of 3 people. How many ways to seat them around a circular table with each party sitting together? Since each party is a single person, so they can be arranged in (3-1)! = 2 ways. But since each person is a different party, the answer should be 2. But if we think of it as 2! * 1! * 1! = 2. So that works. If we have 2 Democrats, 1 Republican, 1 Independent. Then, the number of ways should be (number of blocks -1)! * internal arrangements. The blocks are D (size 2), R (size 1), I (size 1). So number of blocks is 3. (3-1)! = 2. Multiply by internal arrangements: 2! * 1! * 1! = 2*2=4. Let's check manually. Fix the D block. Then arrange R and I. There are 2 ways. Then within D, 2! ways. So total 2*2=4. Which is correct. So that seems to check out.

Therefore, going back, in the original problem, the answer would be 2 * 5! * 5! = 2 * 120 * 120 = 28800. But let me check again with another approach.

Alternatively, sometimes when dealing with circular permutations with groups, you can think of it as linear permutations divided by the number of rotations. Wait, but maybe not. Let me see.

Suppose we first consider the problem as arranging the three blocks in a line. The number of ways would be 3! * 5! * 5! * 1. But since it's a circular table, we have to adjust for rotational symmetry. In linear arrangements, there are 3! ways, but in circular, it's (3-1)! = 2!. Therefore, the answer would be 2! * 5! * 5! * 1. Which is the same as before. So that seems consistent.

But wait, is there a case where the Independent could be considered as part of another block? For example, maybe the problem is considering that the Independent is not part of a party? Wait, the problem says "all the members of each party all sit next to each other." So the parties are Democrats, Republicans, and the Independent. So the Independent is a party of one. Therefore, the answer is as we thought.

But let me think again. Suppose we didn't consider the Independent as a block. Suppose someone thought that since the Independent is not a party with multiple members, they can be placed anywhere. But no, the problem says that all members of each party must sit together. So the Independent, as a single member, is a party, so they have to sit together, which is just themselves. Therefore, they need to be considered as a block. Therefore, the previous reasoning is correct.

Alternatively, maybe there's a different approach. Let me imagine arranging the entire group. Since the Democrats must be together, Republicans must be together, and the Independent must be alone. So, we can think of it as:

First, arrange the Democrats as a block. Then arrange the Republicans as a block. Then the Independent as a single block. Now, arrange these three blocks around the table. For circular arrangements, as before, it's (3-1)! = 2.

But then, the number of ways to arrange within the blocks: Democrats can be arranged in 5! ways, Republicans in 5! ways, and the Independent is just 1 way. So total is 2 * 5! * 5!.

Wait, but maybe the Independent could be placed between the Democrats and Republicans? But since the blocks are arranged around the table, it's accounted for in the circular arrangement. So, for example, the order could be D, R, I or D, I, R or R, D, I, etc., but since it's circular, we fix one block and arrange the others. Hence, 2 ways.

Alternatively, maybe I can think of the entire seating as follows:

Imagine the table has seats labeled 1 to n, but since rotations are considered the same, we can fix one person's position. Let's fix the Independent's position. Then, we have to arrange the Democrats and Republicans. But since Democrats must be together and Republicans must be together, we can think of arranging the two blocks (Democrats and Republicans) around the table relative to the Independent. But since the Independent is fixed, the two blocks can be arranged in 2! ways: either Democrats first then Republicans, or Republicans first then Democrats. Then, within each block, the Democrats can be arranged in 5! ways and Republicans in 5! ways. So total arrangements would be 2! * 5! * 5! = 2 * 120 * 120 = 28800. Which is the same answer. So that seems to confirm it.

But, wait, if I fix the Independent's position, isn't that equivalent to considering circular permutations by fixing a position? Yes. So fixing one position accounts for rotational symmetry, so the rest can be arranged linearly. Therefore, in this case, fixing the Independent, we have two blocks left, which can be arranged in 2! ways. Then multiplied by internal arrangements. That gives the same answer. Therefore, the answer should be 2 * 5! * 5!.

But let me check again for overcounting or undercounting.

Is there any chance that the Independent could be part of another block? For example, if the problem had said that the Independent can be with either Democrats or Republicans, but no, the problem states that all members of each party must sit together. So the Independent is a separate party, so they have to be seated alone. Therefore, the only way for the Independent to be seated is as a single block. Therefore, there is no ambiguity here.

Alternatively, let me think of the problem as first arranging the Democrats, then the Republicans, then the Independent. But in circular arrangements.

But how?

Alternatively, suppose we consider the three blocks D, R, I. In circular permutations, the number of ways to arrange them is (3-1)! = 2. Then, for each block, we have 5! for D, 5! for R, and 1! for I. So 2 * 5! * 5!.

Yes, that seems correct.

Wait, but another thought: what if the Independent is not considered as a block? For example, if we have the Democrats sitting together, Republicans sitting together, and the Independent is somewhere else. But the problem says "all the members of each party all sit next to each other". Since the Independent is a party of one, they have to sit next to themselves, which is trivial. Therefore, the Independent can be placed anywhere, but since the Democrats and Republicans are blocks, the Independent is just an individual. Therefore, the seating arrangement must have the Democrats as a block, Republicans as a block, and the Independent as an individual. Therefore, the three blocks are: D, R, and I.

But if I think about that, the individual I is not a block. Wait, but if we have the three blocks, they are D (size 5), R (size 5), and I (size 1). Therefore, total people are 5 + 5 + 1 = 11. So arranging those three blocks around the table. The number of ways to arrange the blocks is (3-1)! = 2. Multiply by the internal arrangements. So yes, 2 * 5! * 5! * 1! = 2 * 5! * 5!.

But maybe someone could argue that the Independent is not a block, so we need to consider the blocks D and R, and then place the Independent in between? Wait, but the problem says all members of each party must sit together, so if the Independent is not a party, but a single member, then they have to be seated somewhere, but they are not required to be with anyone else. 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. Therefore, the Independent is a party, so they have to sit together. Which is just themselves. Therefore, they have to be considered as a block.

Alternatively, if the problem had said "Democrats, Republicans, and the Independent", but the Independent is not a party, then the answer would be different. But the problem says "all the members of each party all sit next to each other". Since there's an Independent, it's implied that they are part of a party, so they must sit together. Therefore, the answer is indeed 2 * 5! * 5!.

But to be thorough, let me check an alternative approach. Suppose we first consider the entire circular arrangement where Democrats are all together, Republicans are all together, and the Independent is somewhere. How would that work?

Suppose we first treat the Democrats as a single block. Then, the Republicans as a single block. Then, the Independent is a single person. Now, arranging these three entities (D-block, R-block, I) around the table. Since it's a circular table, the number of arrangements is (3-1)! = 2. Then, multiply by the internal arrangements of the D-block (5!), R-block (5!), and the I (1). Therefore, same answer.

Alternatively, if we didn't consider the Independent as a block. Suppose we thought that the Independent can be inserted into the arrangement. But that would be incorrect because if we consider the Democrats and Republicans as blocks, then the Independent is just another entity. But since there are three entities, the number of ways to arrange them is (3-1)!.

Alternatively, someone might think that the Independent can be placed anywhere between the Democrats and Republicans. But since we have three blocks, the Independent is a separate block.

Alternatively, if the problem had said that the Independent is not a party, but just a single member, then we would have two blocks (Democrats and Republicans) and one individual. But since the problem states that all members of each party must sit together, and the Independent is a member of a party (as they are described as "the Independent", implying they are a party of one), so they must be seated together. So that is why we have three blocks.

Another way to think about it: if we didn't have the Independent, and just had Democrats and Republicans, then the number of arrangements would be (2-1)! * 5! * 5! = 1 * 5! * 5! = 14400. But with the Independent, it's adding another block, so we have 3 blocks. Thus (3-1)! * 5! * 5! * 1! = 2 * 14400 = 28800. Which is what we got before.

But let me check with another approach. Suppose we first arrange the Democrats, Republicans, and Independent in a line, then adjust for circular permutations.

In linear arrangements, the number of ways would be: arranging the three blocks (D, R, I) in 3! ways, multiplied by internal arrangements (5! * 5! * 1). However, since it's a circular table, we need to divide by the number of rotations, which is equal to the number of seats, but wait, that might not be straightforward.

Alternatively, for circular arrangements, the formula is (number of linear arrangements) / number of seats. Wait, no. For circular arrangements, the number of distinct arrangements is (number of linear arrangements) divided by the number of seats, because each circular arrangement corresponds to n linear arrangements (one for each rotation). However, in this case, the blocks are of different sizes. Wait, but the total number of seats is 11. However, if we think of arranging the blocks in a circle, the formula is (number of ways to arrange blocks linearly) divided by number of blocks? Wait, no. Let me recall.

In general, for circular arrangements with distinguishable blocks, the number of arrangements is (number of blocks - 1)! So if we have three blocks, it's 2!.

But if we consider the linear arrangements, it's 3! Then, since in circular arrangements, each unique arrangement is counted n times in linear arrangements, but here n is the number of seats? Wait, no. Wait, for circular arrangements of distinct objects, it's (number of objects - 1)! So if the number of objects is 3 (blocks), then it's 2!.

But if we have linear arrangements of the blocks, it's 3! Then, to get circular arrangements, divide by 3 (number of blocks?), but that seems inconsistent with the usual formula. Wait, no, the standard formula is that circular arrangements are (linear arrangements)/n, where n is the number of objects. Wait, if you have n objects in a circle, the number of circular arrangements is n!/n = (n-1)!.

But here, the objects are blocks, and the number of blocks is 3, so the number of circular arrangements would be (3!)/3 = 2, which matches the formula (3-1)! = 2. Therefore, if we think of the linear arrangements as 3! and divide by 3 (number of blocks) to account for rotations, we get 2. So that's consistent. Therefore, the total number of circular arrangements is (number of ways to arrange blocks linearly) divided by number of blocks.

But in this case, the number of ways to arrange the blocks linearly is 3! * 5! * 5! * 1! = 6 * 120 * 120 = 86400. Then, dividing by 3 (number of blocks) gives 86400 / 3 = 28800. Which matches our previous answer. Therefore, this confirms that the answer is 28800.

Therefore, I think that 2 * 5! * 5! is correct.

Wait, so 3! / 3 = 2, so the number of circular arrangements is 2 * 5! * 5!.

Alternatively, another way to think about it is that when you have circular arrangements with blocks, you can fix one block's position to remove rotational symmetry. For example, fix the Democrats block in a specific position. Then arrange the other two blocks (Republicans and Independent) around the table. There are 2! ways to arrange them. Then multiply by the internal arrangements of each block. So, 2! * 5! * 5!.

Yes, exactly. So fixing one block's position gives us the correct count. Therefore, 2 * 5! * 5!.

So, I think that's solid. Therefore, the answer is 2 * 5! * 5!.

Calculating that, 5! is 120, so 120 * 120 = 14400, and 14400 * 2 = 28800. Therefore, the number of ways is 28,800.

But just to be 100% sure, let me check if there is a different interpretation where the answer could be different. For example, could the Independent be considered as part of either Democrats or Republicans? But the problem states that all members of each party must sit together. If the Independent were part of a party, they would have to be seated with that party. However, the problem says "the Independent", implying that they are a separate party. Therefore, they have to be seated alone. Therefore, there is no way they can be merged with Democrats or Republicans. Hence, the original approach is correct.

Another possible confusion: is the Independent considered a separate party, so the total parties are three: Democrats, Republicans, Independent? If so, then yes, they have to be considered as separate blocks. If the problem had said that there are three parties: Democrats, Republicans, and the Independent, then that's three parties. But in reality, the Independent is not a party, but a single member. However, the problem says "all the members of each party all sit next to each other." Since the Independent is a member of a party (the Independent party), they have to sit next to each other. Therefore, the answer is as we calculated.

Alternatively, if the problem had said "Democrats, Republicans, and the Independent is not a party," then the answer would be different. But the problem does not say that. It says "all the members of each party all sit next to each other." Therefore, the Independent is a member of a party (the Independent party), so they need to be seated together. Therefore, the answer is 2 * 5! * 5!.

Thus, I think I'm confident that the answer is 28800. Therefore, the final answer is $\boxed{28800}$.

**Final Answer**
\boxed{28800}

To solve the problem, we're asked to find the number of distinct seating arrangements around a **circular table** for a Senate committee consisting of:

- 5 Democrats
- 5 Republicans
- 1 Independent

With the condition that **all members of each party must sit next to each other**. Furthermore, two seatings are considered equivalent if one can be obtained by **rotating** the other.

---

### Step 1: Understand the Structure

Given the condition that members of each party must sit together, we can **treat the members of each party as a single block**.

So we have three blocks:

- **D Block (Democrats)**: size 5
- **R Block (Republicans)**: size 5
- **I Block (Independent)**: size 1

These blocks must be arranged around the table **in some order**, while their internal members can be permuted.

---

### Step 2: Circular Permutations of Blocks

In circular arrangements of **distinct** objects, the number of unique arrangements is given by:

$$
(\text{number of blocks} - 1)!
$$

Here, we have 3 blocks, so the number of distinct circular arrangements of the blocks is:

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

So, there are **2** distinct ways to arrange the block order around the table.

---

### Step 3: Internal Permutations Within Each Block

Each block contains members that can be permuted:

- **Democrats block**: 5 people → $ 5! $ permutations
- **Republicans block**: 5 people → $ 5! $ permutations
- **Independent block**: 1 person → $ 1! $ permutation (only 1 way)

Therefore, the total number of internal arrangements is:

$$
5! \times 5! \times 1! = 120 \times 120 \times 1 = 14400
$$

---

### Step 4: Total Number of Arrangements

Multiply the number of circular block arrangements by the internal permutations of each block:

$$
2! \times 5! \times 5! = 2 \times 120 \times 120 = 28800
$$

---

### 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 about how to approach this.

First, circular permutations can be tricky because rotations are considered the same. Normally, for n people sitting 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 the Independent. However, the Independent is only one person. So does that mean the Independent can be considered as a block of size 1? Or does the problem require that all members of each party are seated together, so the Independent is a party of one? Let me check the problem statement again: "all the members of each party all sit next to each other." Since there is only one Independent, that person is already sitting alone, so technically, the condition is satisfied because the single member is trivially sitting next to themselves? Maybe.

So, if we think of the three parties as three blocks: D (Democrats), R (Republicans), and I (Independent). But since the Independent is just one person, their block is just one person. So, how many ways can we arrange these three blocks around the table?

But since it's a circular table, arranging blocks around a circular table. Normally, for circular arrangements of k blocks, it's (k-1)! ways. But here, the blocks have internal arrangements. So, for each block, we need to consider the number of ways to arrange the members within the block.

So, first, arrange the three blocks around the table. Since rotations are considered the same, it's (3-1)! = 2! = 2 ways. Then, for each block, multiply by the number of internal arrangements.

But wait, the Independent is a single person, so there's only 1 way to arrange them. The Democrats are 5 people, so they can be arranged in 5! ways. Similarly, the Republicans are 5 people, so 5! ways. So total number would be 2! * 5! * 5! * 1?

Wait, that seems too small. Let me check again. Wait, if we have three blocks: D, R, I. Since it's a circular table, arranging three blocks would be (3-1)! = 2 ways. Then multiply by the permutations within each block. So that would be 2 * 5! * 5! * 1. But is that correct?

But wait, let me think again. Suppose we fix the position of one block to account for the circular table. For example, fix the Democrats block in a certain position. Then arrange the other two blocks (Republicans and Independent) around the table. Since there are 2! ways to arrange the remaining two blocks. Then, within the Democrats block, 5! arrangements, Republicans block 5! arrangements, and the Independent is fixed. So total is 2! * 5! * 5!.

But why does fixing the Democrats block account for the circular rotation? Because in circular permutations, fixing one position removes rotational symmetry. So yes, that seems right. So the answer would be 2 * 5! * 5!.

Wait, but let me check if there's another way to think about it. Alternatively, sometimes when dealing with circular permutations with blocks, the formula is (number of blocks - 1)! multiplied by the product of the internal arrangements. So here, the number of blocks is 3, so (3-1)! * 5! * 5! * 1. So that's 2! * 5! * 5! = 2 * 120 * 120 = 2 * 14400 = 28800. Is that the answer? Hmm, but maybe I'm missing something?

Wait, let me consider the Independent. Since the Independent is a single person, but is there a different way they can be arranged? Or is the Independent considered as a block of size 1? Well, if we treat them as a block, then arranging the three blocks (D, R, I) around the table, which is (3-1)! = 2 ways. Then multiply by the internal arrangements. For D: 5!, R: 5!, I: 1! = 1. So total is 2 * 5! * 5! = 28800. So maybe that is correct?

But let me think again. What if the Independent is considered as a separate block, so the three blocks are D, R, and I. But in circular arrangements, if you have three blocks, how many ways are there to arrange them? It's (3-1)! = 2. But if the blocks are different, then that's correct. So yes, two ways to arrange the blocks.

But wait, suppose that the Independent is not a block? Wait, no. Because the problem says that all members of each party must sit next to each other. Since the Independent is a party by themselves, they have to sit next to themselves. Which is trivially true. So the Independent is a block of size 1.

But let me check if there's another interpretation. Suppose that the problem requires that each party (Democrats, Republicans, and the Independent) are all seated together. However, the Independent is only one person, so they are already seated alone. Therefore, the only way for the Independent to be seated together is if they are seated alone. Therefore, the problem is equivalent to arranging Democrats, Republicans, and Independent as blocks. So the answer should be (number of ways to arrange the blocks) multiplied by internal arrangements.

But let me check another way. Suppose we think of the entire seating as three blocks: D, R, I. Since it's a circular table, the number of ways to arrange these blocks is (3-1)! = 2. Then, for each block, the number of arrangements is 5! for Democrats, 5! for Republicans, and 1! for the Independent. So total is 2 * 5! * 5!.

But perhaps there is a mistake here. Let me think. What if we didn't fix the blocks? For example, if there are three blocks, the number of circular permutations is (number of blocks - 1)! So 2! = 2. So that seems correct. Then multiplied by internal permutations. So 2 * 5! * 5!.

But let me see if I can think of an example with smaller numbers. Let's say there are 1 Democrat, 1 Republican, and 1 Independent. So total of 3 people. How many ways to seat them around a circular table with each party sitting together? Since each party is a single person, so they can be arranged in (3-1)! = 2 ways. But since each person is a different party, the answer should be 2. But if we think of it as 2! * 1! * 1! = 2. So that works. If we have 2 Democrats, 1 Republican, 1 Independent. Then, the number of ways should be (number of blocks -1)! * internal arrangements. The blocks are D (size 2), R (size 1), I (size 1). So number of blocks is 3. (3-1)! = 2. Multiply by internal arrangements: 2! * 1! * 1! = 2*2=4. Let's check manually. Fix the D block. Then arrange R and I. There are 2 ways. Then within D, 2! ways. So total 2*2=4. Which is correct. So that seems to check out.

Therefore, going back, in the original problem, the answer would be 2 * 5! * 5! = 2 * 120 * 120 = 28800. But let me check again with another approach.

Alternatively, sometimes when dealing with circular permutations with groups, you can think of it as linear permutations divided by the number of rotations. Wait, but maybe not. Let me see.

Suppose we first consider the problem as arranging the three blocks in a line. The number of ways would be 3! * 5! * 5! * 1. But since it's a circular table, we have to adjust for rotational symmetry. In linear arrangements, there are 3! ways, but in circular, it's (3-1)! = 2!. Therefore, the answer would be 2! * 5! * 5! * 1. Which is the same as before. So that seems consistent.

But wait, is there a case where the Independent could be considered as part of another block? For example, maybe the problem is considering that the Independent is not part of a party? Wait, the problem says "all the members of each party all sit next to each other." So the parties are Democrats, Republicans, and the Independent. So the Independent is a party of one. Therefore, the answer is as we thought.

But let me think again. Suppose we didn't consider the Independent as a block. Suppose someone thought that since the Independent is not a party with multiple members, they can be placed anywhere. But no, the problem says that all members of each party must sit together. So the Independent, as a single member, is a party, so they have to sit together, which is just themselves. Therefore, they need to be considered as a block. Therefore, the previous reasoning is correct.

Alternatively, maybe there's a different approach. Let me imagine arranging the entire group. Since the Democrats must be together, Republicans must be together, and the Independent must be alone. So, we can think of it as:

First, arrange the Democrats as a block. Then arrange the Republicans as a block. Then the Independent as a single block. Now, arrange these three blocks around the table. For circular arrangements, as before, it's (3-1)! = 2.

But then, the number of ways to arrange within the blocks: Democrats can be arranged in 5! ways, Republicans in 5! ways, and the Independent is just 1 way. So total is 2 * 5! * 5!.

Wait, but maybe the Independent could be placed between the Democrats and Republicans? But since the blocks are arranged around the table, it's accounted for in the circular arrangement. So, for example, the order could be D, R, I or D, I, R or R, D, I, etc., but since it's circular, we fix one block and arrange the others. Hence, 2 ways.

Alternatively, maybe I can think of the entire seating as follows:

Imagine the table has seats labeled 1 to n, but since rotations are considered the same, we can fix one person's position. Let's fix the Independent's position. Then, we have to arrange the Democrats and Republicans. But since Democrats must be together and Republicans must be together, we can think of arranging the two blocks (Democrats and Republicans) around the table relative to the Independent. But since the Independent is fixed, the two blocks can be arranged in 2! ways: either Democrats first then Republicans, or Republicans first then Democrats. Then, within each block, the Democrats can be arranged in 5! ways and Republicans in 5! ways. So total arrangements would be 2! * 5! * 5! = 2 * 120 * 120 = 28800. Which is the same answer. So that seems to confirm it.

But, wait, if I fix the Independent's position, isn't that equivalent to considering circular permutations by fixing a position? Yes. So fixing one position accounts for rotational symmetry, so the rest can be arranged linearly. Therefore, in this case, fixing the Independent, we have two blocks left, which can be arranged in 2! ways. Then multiplied by internal arrangements. That gives the same answer. Therefore, the answer should be 2 * 5! * 5!.

But let me check again for overcounting or undercounting.

Is there any chance that the Independent could be part of another block? For example, if the problem had said that the Independent can be with either Democrats or Republicans, but no, the problem states that all members of each party must sit together. So the Independent is a separate party, so they have to be seated alone. Therefore, the only way for the Independent to be seated is as a single block. Therefore, there is no ambiguity here.

Alternatively, let me think of the problem as first arranging the Democrats, then the Republicans, then the Independent. But in circular arrangements.

But how?

Alternatively, suppose we consider the three blocks D, R, I. In circular permutations, the number of ways to arrange them is (3-1)! = 2. Then, for each block, we have 5! for D, 5! for R, and 1! for I. So 2 * 5! * 5!.

Yes, that seems correct.

Wait, but another thought: what if the Independent is not considered as a block? For example, if we have the Democrats sitting together, Republicans sitting together, and the Independent is somewhere else. But the problem says "all the members of each party all sit next to each other". Since the Independent is a party of one, they have to sit next to themselves, which is trivial. Therefore, the Independent can be placed anywhere, but since the Democrats and Republicans are blocks, the Independent is just an individual. Therefore, the seating arrangement must have the Democrats as a block, Republicans as a block, and the Independent as an individual. Therefore, the three blocks are: D, R, and I.

But if I think about that, the individual I is not a block. Wait, but if we have the three blocks, they are D (size 5), R (size 5), and I (size 1). Therefore, total people are 5 + 5 + 1 = 11. So arranging those three blocks around the table. The number of ways to arrange the blocks is (3-1)! = 2. Multiply by the internal arrangements. So yes, 2 * 5! * 5! * 1! = 2 * 5! * 5!.

But maybe someone could argue that the Independent is not a block, so we need to consider the blocks D and R, and then place the Independent in between? Wait, but the problem says all members of each party must sit together, so if the Independent is not a party, but a single member, then they have to be seated somewhere, but they are not required to be with anyone else. 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. Therefore, the Independent is a party, so they have to sit together. Which is just themselves. Therefore, they have to be considered as a block.

Alternatively, if the problem had said "Democrats, Republicans, and the Independent", but the Independent is not a party, then the answer would be different. But the problem says "all the members of each party all sit next to each other". Since there's an Independent, it's implied that they are part of a party, so they must sit together. Therefore, the answer is indeed 2 * 5! * 5!.

But to be thorough, let me check an alternative approach. Suppose we first consider the entire circular arrangement where Democrats are all together, Republicans are all together, and the Independent is somewhere. How would that work?

Suppose we first treat the Democrats as a single block. Then, the Republicans as a single block. Then, the Independent is a single person. Now, arranging these three entities (D-block, R-block, I) around the table. Since it's a circular table, the number of arrangements is (3-1)! = 2. Then, multiply by the internal arrangements of the D-block (5!), R-block (5!), and the I (1). Therefore, same answer.

Alternatively, if we didn't consider the Independent as a block. Suppose we thought that the Independent can be inserted into the arrangement. But that would be incorrect because if we consider the Democrats and Republicans as blocks, then the Independent is just another entity. But since there are three entities, the number of ways to arrange them is (3-1)!.

Alternatively, someone might think that the Independent can be placed anywhere between the Democrats and Republicans. But since we have three blocks, the Independent is a separate block.

Alternatively, if the problem had said that the Independent is not a party, but just a single member, then we would have two blocks (Democrats and Republicans) and one individual. But since the problem states that all members of each party must sit together, and the Independent is a member of a party (as they are described as "the Independent", implying they are a party of one), so they must be seated together. So that is why we have three blocks.

Another way to think about it: if we didn't have the Independent, and just had Democrats and Republicans, then the number of arrangements would be (2-1)! * 5! * 5! = 1 * 5! * 5! = 14400. But with the Independent, it's adding another block, so we have 3 blocks. Thus (3-1)! * 5! * 5! * 1! = 2 * 14400 = 28800. Which is what we got before.

But let me check with another approach. Suppose we first arrange the Democrats, Republicans, and Independent in a line, then adjust for circular permutations.

In linear arrangements, the number of ways would be: arranging the three blocks (D, R, I) in 3! ways, multiplied by internal arrangements (5! * 5! * 1). However, since it's a circular table, we need to divide by the number of rotations, which is equal to the number of seats, but wait, that might not be straightforward.

Alternatively, for circular arrangements, the formula is (number of linear arrangements) / number of seats. Wait, no. For circular arrangements, the number of distinct arrangements is (number of linear arrangements) divided by the number of seats, because each circular arrangement corresponds to n linear arrangements (one for each rotation). However, in this case, the blocks are of different sizes. Wait, but the total number of seats is 11. However, if we think of arranging the blocks in a circle, the formula is (number of ways to arrange blocks linearly) divided by number of blocks? Wait, no. Let me recall.

In general, for circular arrangements with distinguishable blocks, the number of arrangements is (number of blocks - 1)! So if we have three blocks, it's 2!.

But if we consider the linear arrangements, it's 3! Then, since in circular arrangements, each unique arrangement is counted n times in linear arrangements, but here n is the number of seats? Wait, no. Wait, for circular arrangements of distinct objects, it's (number of objects - 1)! So if the number of objects is 3 (blocks), then it's 2!.

But if we have linear arrangements of the blocks, it's 3! Then, to get circular arrangements, divide by 3 (number of blocks?), but that seems inconsistent with the usual formula. Wait, no, the standard formula is that circular arrangements are (linear arrangements)/n, where n is the number of objects. Wait, if you have n objects in a circle, the number of circular arrangements is n!/n = (n-1)!.

But here, the objects are blocks, and the number of blocks is 3, so the number of circular arrangements would be (3!)/3 = 2, which matches the formula (3-1)! = 2. Therefore, if we think of the linear arrangements as 3! and divide by 3 (number of blocks) to account for rotations, we get 2. So that's consistent. Therefore, the total number of circular arrangements is (number of ways to arrange blocks linearly) divided by number of blocks.

But in this case, the number of ways to arrange the blocks linearly is 3! * 5! * 5! * 1! = 6 * 120 * 120 = 86400. Then, dividing by 3 (number of blocks) gives 86400 / 3 = 28800. Which matches our previous answer. Therefore, this confirms that the answer is 28800.

Therefore, I think that 2 * 5! * 5! is correct.

Wait, so 3! / 3 = 2, so the number of circular arrangements is 2 * 5! * 5!.

Alternatively, another way to think about it is that when you have circular arrangements with blocks, you can fix one block's position to remove rotational symmetry. For example, fix the Democrats block in a specific position. Then arrange the other two blocks (Republicans and Independent) around the table. There are 2! ways to arrange them. Then multiply by the internal arrangements of each block. So, 2! * 5! * 5!.

Yes, exactly. So fixing one block's position gives us the correct count. Therefore, 2 * 5! * 5!.

So, I think that's solid. Therefore, the answer is 2 * 5! * 5!.

Calculating that, 5! is 120, so 120 * 120 = 14400, and 14400 * 2 = 28800. Therefore, the number of ways is 28,800.

But just to be 100% sure, let me check if there is a different interpretation where the answer could be different. For example, could the Independent be considered as part of either Democrats or Republicans? But the problem states that all members of each party must sit together. If the Independent were part of a party, they would have to be seated with that party. However, the problem says "the Independent", implying that they are a separate party. Therefore, they have to be seated alone. Therefore, there is no way they can be merged with Democrats or Republicans. Hence, the original approach is correct.

Another possible confusion: is the Independent considered a separate party, so the total parties are three: Democrats, Republicans, Independent? If so, then yes, they have to be considered as separate blocks. If the problem had said that there are three parties: Democrats, Republicans, and the Independent, then that's three parties. But in reality, the Independent is not a party, but a single member. However, the problem says "all the members of each party all sit next to each other." Since the Independent is a member of a party (the Independent party), they have to sit next to each other. Therefore, the answer is as we calculated.

Alternatively, if the problem had said "Democrats, Republicans, and the Independent is not a party," then the answer would be different. But the problem does not say that. It says "all the members of each party all sit next to each other." Therefore, the Independent is a member of a party (the Independent party), so they need to be seated together. Therefore, the answer is 2 * 5! * 5!.

Thus, I think I'm confident that the answer is 28800. Therefore, the final answer is $\boxed{28800}$.

**Final Answer**
\boxed{28800}

To solve the problem, we're asked to find the number of distinct seating arrangements around a **circular table** for a Senate committee consisting of:

- 5 Democrats
- 5 Republicans
- 1 Independent

With the condition that **all members of each party must sit next to each other**. Furthermore, two seatings are considered equivalent if one can be obtained by **rotating** the other.

---

### Step 1: Understand the Structure

Given the condition that members of each party must sit together, we can **treat the members of each party as a single block**.

So we have three blocks:

- **D Block (Democrats)**: size 5
- **R Block (Republicans)**: size 5
- **I Block (Independent)**: size 1

These blocks must be arranged around the table **in some order**, while their internal members can be permuted.

---

### Step 2: Circular Permutations of Blocks

In circular arrangements of **distinct** objects, the number of unique arrangements is given by:

$$
(\text{number of blocks} - 1)!
$$

Here, we have 3 blocks, so the number of distinct circular arrangements of the blocks is:

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

So, there are **2** distinct ways to arrange the block order around the table.

---

### Step 3: Internal Permutations Within Each Block

Each block contains members that can be permuted:

- **Democrats block**: 5 people → $ 5! $ permutations
- **Republicans block**: 5 people → $ 5! $ permutations
- **Independent block**: 1 person → $ 1! $ permutation (only 1 way)

Therefore, the total number of internal arrangements is:

$$
5! \times 5! \times 1! = 120 \times 120 \times 1 = 14400
$$

---

### Step 4: Total Number of Arrangements

Multiply the number of circular block arrangements by the internal permutations of each block:

$$
2! \times 5! \times 5! = 2 \times 120 \times 120 = 28800
$$

---

### 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 show up in problems involving complex numbers or trigonometric identities. Maybe there's a pattern or recurrence relation here. Let me recall that if we have $ z + 1/z = k $, then powers of $ z $ can be related to each other through some recursive formula.

Let me try to define $ a_n = z^n + \frac{1}{z^n} $. Then, maybe I can find a recurrence relation for $ a_n $. Let me check for small n.

For $ n = 1 $, we have $ a_1 = z + 1/z = \frac{1 + \sqrt{5}}{2} $, which is given.

What about $ a_2 $? Let's compute $ a_2 = z^2 + 1/z^2 $. There's a formula for that in terms of $ a_1 $:

$ a_2 = (z + 1/z)^2 - 2 = a_1^2 - 2 $.

Similarly, for $ a_3 $, maybe there's a similar relation. Let me think. If I multiply $ a_1 $ and $ a_2 $, perhaps?

Wait, $ a_1 \times a_2 = (z + 1/z)(z^2 + 1/z^2) = z^3 + z^{-1} + z + z^{-3} $. That is $ a_3 + a_1 $. So $ a_1 a_2 = a_3 + a_1 $, which gives $ a_3 = a_1 a_2 - a_1 $.

Alternatively, there's a general recurrence relation for $ a_n $. Let me recall:

If $ a_{n+1} = a_1 a_n - a_{n-1} $.

Yes, that seems familiar. Let me verify this for n=2:

$ a_2 = a_1 a_1 - a_0 $. Wait, but what is $ a_0 $? Since $ a_0 = z^0 + 1/z^0 = 1 + 1 = 2 $.

So indeed, for n=1, $ a_1 = z + 1/z $, and for n=2, $ a_2 = a_1 a_1 - a_0 = (a_1)^2 - 2 $, which matches the earlier formula.

So the recurrence relation seems to be $ a_{n} = a_1 a_{n-1} - a_{n-2} $.

Therefore, if I can compute $ a_n $ up to n=85 using this recurrence, then I can find the desired value. However, computing up to 85 manually would be tedious. Maybe there's a pattern or periodicity?

Alternatively, perhaps the sequence $ a_n $ follows some periodic behavior? Let me check with the given value of $ a_1 $, which is the golden ratio, $ \phi = \frac{1 + \sqrt{5}}{2} $.

Let me compute the first few terms to see if a pattern emerges.

First, let me compute $ a_1 = \phi $.

Compute $ a_2 $:

$ a_2 = \phi^2 - 2 $.

But since $ \phi $ satisfies the equation $ \phi^2 = \phi + 1 $, so substituting:

$ a_2 = (\phi + 1) - 2 = \phi - 1 $.

Wait, $ \phi - 1 $ is actually $ \frac{1}{\phi} $. Because $ \phi = \frac{1 + \sqrt{5}}{2} $, then $ 1/\phi = \frac{\sqrt{5} - 1}{2} $, which is equal to $ \phi - 1 $. Let me confirm:

$ \phi - 1 = \frac{1 + \sqrt{5}}{2} - 1 = \frac{-1 + \sqrt{5}}{2} = \frac{\sqrt{5} - 1}{2} $, and $ 1/\phi = \frac{2}{1 + \sqrt{5}} \times \frac{\sqrt{5} - 1}{\sqrt{5} - 1} = \frac{2(\sqrt{5} - 1)}{5 - 1} = \frac{2(\sqrt{5} - 1)}{4} = \frac{\sqrt{5} - 1}{2} $. Yes, they are equal. Therefore, $ a_2 = 1/\phi $.

Now, moving on to $ a_3 $. Using the recurrence $ a_3 = a_1 a_2 - a_1 $. Wait, no, the recurrence is $ a_n = a_1 a_{n-1} - a_{n - 2} $. Therefore, for $ n = 3 $, $ a_3 = a_1 a_2 - a_1 $. Wait, $ a_1 a_2 - a_0 $? Wait, no. Let me check again.

Wait, general recurrence is $ a_n = a_1 a_{n-1} - a_{n-2} $. For n=2, it would be $ a_2 = a_1 a_1 - a_0 $, which is $ a_1^2 - 2 $. For n=3, it's $ a_3 = a_1 a_2 - a_1 $. Wait, but $ a_1 a_2 - a_1 $? Wait, that seems inconsistent. Let me check with the previous multiplication:

Earlier I had $ a_1 a_2 = a_3 + a_1 $, so rearranged, $ a_3 = a_1 a_2 - a_1 $. Therefore, for n=3, $ a_3 = a_1 a_2 - a_1 $. But according to the general recurrence, $ a_3 = a_1 a_2 - a_1 $, yes. So that works.

Therefore, $ a_3 = \phi \times (1/\phi) - \phi = 1 - \phi $. Wait, $ 1 - \phi $ is negative? Let me compute numerically:

$ \phi \approx 1.618 $, so $ 1 - \phi \approx -0.618 $, which is $ -1/\phi $. Let me verify:

Since $ \phi - 1 = 1/\phi \approx 0.618 $, so $ 1 - \phi = -1/\phi $. Therefore, $ a_3 = -1/\phi $.

Now let me compute $ a_4 $ using the recurrence $ a_4 = a_1 a_3 - a_2 $.

So substituting, $ a_4 = \phi \times (-1/\phi) - (1/\phi) = -1 - 1/\phi $. Wait, that seems messy. Let me check with another approach. Maybe using the identity $ a_4 = (z^2 + 1/z^2)^2 - 2 = a_2^2 - 2 $.

Let me compute that:

$ a_2 = 1/\phi $, so $ a_2^2 - 2 = (1/\phi)^2 - 2 $. But $ 1/\phi^2 = (\phi - 1)^2 = \phi^2 - 2\phi + 1 $. Wait, let me compute $ 1/\phi^2 $:

Since $ \phi^2 = \phi + 1 $, so $ 1/\phi^2 = 1/(\phi + 1) $. Alternatively, maybe another approach.

Alternatively, maybe I can use the recurrence relation. Let me proceed with the recurrence:

$ a_4 = a_1 a_3 - a_2 = \phi \times (-1/\phi) - (1/\phi) = -1 - 1/\phi $.

Wait, but maybe that's correct? Let me compute numerically:

$ \phi \approx 1.618 $, so $ -1 - 1/\phi \approx -1 - 0.618 = -1.618 $, which is -phi. Wait, so $ a_4 = -\phi $.

Alternatively, let me check using the other method: $ a_4 = a_2^2 - 2 $.

Since $ a_2 = 1/\phi \approx 0.618 $, then $ a_2^2 \approx 0.618^2 \approx 0.618 * 0.618. Let me compute:

$ 0.618^2 = (approx) 0.618 * 0.618. Let me recall that $ 1/\phi^2 = (\phi - 1)^2 = \phi^2 - 2\phi + 1. But since $ \phi^2 = \phi + 1, substitute: $ 1/\phi^2 = (\phi + 1) - 2\phi + 1 = -\phi + 2. Therefore, $ a_2^2 = 1/\phi^2 = 2 - \phi $. Hence, $ a_4 = (2 - \phi) - 2 = -\phi $. Yes! So that matches the previous result.

So $ a_4 = -\phi $.

Now $ a_5 = a_1 a_4 - a_3 = \phi \times (-\phi) - (-1/\phi) = -\phi^2 + 1/\phi $.

But $ \phi^2 = \phi + 1 $, so substitute:

$ -(\phi + 1) + 1/\phi = -\phi -1 + 1/\phi $. But earlier we know that $ 1/\phi = \phi - 1 $. Therefore, substituting:

$ -\phi -1 + (\phi - 1) = (-\phi + \phi) + (-1 -1) = 0 - 2 = -2 $.

So $ a_5 = -2 $.

Interesting. Let's compute $ a_6 $ using the recurrence relation:

$ a_6 = a_1 a_5 - a_4 = \phi \times (-2) - (-\phi) = -2\phi + \phi = -\phi $.

Wait, so $ a_6 = -\phi $. But earlier, $ a_4 = -\phi $. Let me check with another method. Maybe $ a_6 = a_2 a_5 - a_4 $? Wait, no, the recurrence is $ a_n = a_1 a_{n-1} - a_{n-2} $. So for n=6: $ a_6 = a_1 a_5 - a_4 $.

Which is as above. So $ a_6 = -\phi $.

Now $ a_7 = a_1 a_6 - a_5 = \phi \times (-\phi) - (-2) = -\phi^2 + 2 $.

Again, $ \phi^2 = \phi + 1 $, so substituting: $ -(\phi + 1) + 2 = -\phi -1 + 2 = -\phi + 1 $.

Which is $ 1 - \phi = -1/\phi $. Therefore, $ a_7 = -1/\phi $.

Wait, earlier $ a_3 = -1/\phi $, so $ a_7 = a_3 $. Interesting.

Then $ a_8 = a_1 a_7 - a_6 = \phi \times (-1/\phi) - (-\phi) = -1 + \phi $.

Which is $ \phi - 1 = 1/\phi $. So $ a_8 = 1/\phi $, which is $ a_2 $.

Similarly, $ a_9 = a_1 a_8 - a_7 = \phi \times (1/\phi) - (-1/\phi) = 1 + 1/\phi $.

Which is $ 1 + 1/\phi $. But since $ 1 + 1/\phi = 1 + (\phi - 1) = \phi $. Therefore, $ a_9 = \phi $, which is $ a_1 $.

Wait a minute! So let me list the terms I have so far:

n:  0    1       2         3          4         5         6         7         8         9
a_n: 2  φ    1/φ    -1/φ     -φ        -2       -φ      -1/φ     1/φ      φ

So, looking at n=1: φ, n=9: φ. Then n=2: 1/φ, n=8: 1/φ. Then n=3: -1/φ, n=7: -1/φ. n=4: -φ, n=6: -φ. n=5: -2.

So seems like the sequence is periodic with period 8? Wait, from n=1 to n=9, which is 8 steps, and it's repeating? Let me check:

If a_1 = φ, then a_9 = φ. Then, if the sequence repeats every 8 terms, then a_1 = a_9 = a_17 = ... So, to compute a_85, perhaps we can find 85 modulo 8? Wait, but need to check the period.

Wait, but let's check n=9 is a_9 = φ = a_1, n=10 would be a_10 = a_1 * a_9 - a_8 = φ*φ - 1/φ. Let me compute that.

But φ^2 = φ + 1, so a_10 = φ + 1 - 1/φ. Since 1/φ = φ -1, then substituting:

a_10 = φ + 1 - (φ -1) = φ + 1 - φ + 1 = 2. Which is a_0. Then a_10 = 2, which is a_0.

Then a_11 = a_1 a_10 - a_9 = φ * 2 - φ = 2φ - φ = φ, which is a_1. Therefore, the sequence is periodic with period 10? Wait, but from a_0 to a_10, we have a_0 = 2, then a_1 = φ, ..., a_10 = 2. So the period is 10? But earlier from a_1 to a_9 is φ, and then a_10 = 2. Wait, but if the period is 10, then the sequence would repeat every 10 terms. But let me check:

If a_1 = a_11, and a_2 = a_12, etc. But earlier, a_10 = 2 = a_0. So the sequence {a_n} has period 10? Let me check n=10: 2, which is a_0. Then n=11: φ, which is a_1. So yes, the sequence is periodic with period 10 starting from n=0? So the period is 10.

But then if that's the case, then the sequence for n ≥ 0 is:

n:  0: 2

n:  1: φ

n:  2: 1/φ

n:  3: -1/φ

n:  4: -φ

n:  5: -2

n:  6: -φ

n:  7: -1/φ

n:  8: 1/φ

n:  9: φ

n: 10: 2

So yes, the cycle is 10. Therefore, the period is 10. Therefore, if I can compute n modulo 10, then I can find the value of a_n.

But wait, let me confirm with n=5: a_5 = -2, n=15 would be same as n=5? Let me compute a_15. But maybe it's faster to check the period.

Alternatively, since the recurrence relation is linear and homogeneous, the sequence is periodic if the characteristic equation has roots that are roots of unity or something similar. But given that we have this periodicity of 10, perhaps.

But in any case, given that the sequence has period 10, starting from n=0. So to find a_85, since 85 divided by 10 is 8 with remainder 5, because 10*8=80, 85-80=5. Therefore, a_85 = a_5 = -2.

Wait, so according to this, the answer is -2. But let me check again with the earlier terms.

Wait, let me check with n=5: a_5=-2, then n=15 would be a_15=a_5=-2, and so on. Therefore, since 85=10*8 +5, so a_85=a_5=-2. But let me check with another approach to confirm.

Alternatively, maybe there's a mistake in my period? Let me check for n=10: a_10=2, which is a_0. Then a_11=a_1. So the period is 10. So the sequence repeats every 10 terms. Therefore, 85 divided by 10 is 8 with remainder 5, so a_85 = a_5 = -2.

But wait, let me check for n=5 and n=15. Let me compute a_10 through recurrence:

We had up to n=9: a_9 = φ, then a_10 = a_1 * a_9 - a_8 = φ*φ - 1/φ = φ^2 - 1/φ. Since φ^2 = φ + 1, this becomes φ +1 - 1/φ. Since 1/φ = φ -1, substituting:

φ +1 - (φ -1) = φ +1 - φ +1 = 2. So a_10 = 2.

Then a_11 = a_1 * a_10 - a_9 = φ * 2 - φ = 2φ - φ = φ. Which is a_1.

Similarly, a_12 = a_1 * a_11 - a_10 = φ * φ - 2 = φ^2 - 2 = (φ +1) - 2 = φ -1 = 1/φ = a_2.

Therefore, the period is indeed 10. Therefore, the sequence repeats every 10 terms. Therefore, since 85 mod 10 is 5, then a_85 = a_5 = -2.

But let me check another term to be sure. Let's compute a_15. Since 15 mod 10 = 5, so a_15 = a_5 = -2. Let me compute a_15 via recurrence:

We can compute up to a_10=2, a_11=φ, a_12=1/φ, a_13= -1/φ, a_14=-φ, a_15=-2. Yes! So that's correct. Therefore, the pattern holds.

Therefore, answer is -2.

But let me check again if the initial assumption is correct. The problem says z is a complex number such that z + 1/z equals phi. But does that uniquely determine z? Or are there multiple z's? However, regardless of the value of z, the expression z^n + 1/z^n is determined by the recurrence relation, so even though there may be multiple z's, the value of a_n would be same.

Alternatively, if z is on the unit circle, then z = e^{iθ}, and z + 1/z = 2 cosθ. In this case, 2 cosθ = phi. But phi is approx 1.618, which is greater than 2? Wait, no! Wait 2 cosθ can't be more than 2. Wait, phi is approx 1.618, which is less than 2, so that's okay.

Wait, cosθ = phi / 2 ≈ 0.809, which is valid because cos(36°) ≈ 0.809, which is correct. Because phi = 2 cos(36°). So θ = 36° or θ = -36°, etc. Therefore, z could be e^{iπ/5} or e^{-iπ/5}, or other multiples.

But regardless, the expression z^n + 1/z^n would be 2 cos(nθ). Therefore, since θ = π/5 or -π/5, so for example, if θ = π/5, then a_n = 2 cos(nπ/5).

Therefore, let's check for n=1: 2 cos(π/5) = 2*(sqrt(5)+1)/4 = (sqrt(5)+1)/2 = phi. Correct.

Similarly, n=5: 2 cos(5π/5) = 2 cos(π) = -2. Which matches our earlier result of a_5=-2.

Similarly, n=85: 2 cos(85π/5) = 2 cos(17π). Since cos(17π) = cos(π + 16π) = cos(π) = -1. Therefore, 2*(-1) = -2. Which matches. Therefore, regardless of the approach, the answer is -2.

Therefore, even if I consider θ = π/5, then 85θ = 85π/5 = 17π, which is equivalent to π modulo 2π (since 17π = 8*2π + π). Therefore, cos(17π) = cos(π) = -1, so 2 cos(17π) = -2.

Therefore, answer is -2.

Hence, I can confirm that the answer is \boxed{-2}.

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

We are given the equation:

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

Let us define a sequence $\{a_n\}$ as follows:

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

From this definition, we can derive a recurrence relation for $a_n$ based on the identity:

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

This is a second-order linear recurrence relation, and it is satisfied by any number $z$ for which $z + 1/z = a_1$, as long as $z \ne 0$.

---

### Step 1: Determine the Initial Terms

We are given:

- $a_1 = \phi = \frac{1 + \sqrt{5}}{2}$
- $a_0 = z^0 + \frac{1}{z^0} = 1 + 1 = 2$

Using the recurrence, we compute:

- $a_2 = a_1^2 - 2 = \phi^2 - 2 = (\phi + 1) - 2 = \phi - 1 = \frac{1}{\phi}$
- $a_3 = a_1 a_2 - a_1 = \phi \cdot \frac{1}{\phi} - \phi = 1 - \phi = -\frac{1}{\phi}$
- $a_4 = a_1 a_3 - a_2 = \phi \cdot \left(-\frac{1}{\phi}\right) - \frac{1}{\phi} = -1 - \frac{1}{\phi} = -\phi$
- $a_5 = a_1 a_4 - a_3 = \phi \cdot (-\phi) - \left(-\frac{1}{\phi}\right) = -\phi^2 + \frac{1}{\phi} = -(\phi + 1) + \phi - 1 = -2$

---

### Step 2: Identify the Periodicity

Continuing the computation using the recurrence, we eventually find that the sequence $\{a_n\}$ is **periodic with period 10**, as follows:

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

So the sequence repeats every 10 terms:

$$
a_{n+10} = a_n
$$

---

### Step 3: Compute $a_{85}$

We compute the remainder of $85 \mod 10$:

$$
85 \div 10 = 8 \text{ remainder } 5 \Rightarrow 85 \equiv 5 \pmod{10}
$$

Therefore:

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

---

### Final Answer

$$
\boxed{-2}
$$

[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 show up in problems involving complex numbers or trigonometric identities. Maybe there's a pattern or recurrence relation here. Let me recall that if we have $ z + 1/z = k $, then powers of $ z $ can be related to each other through some recursive formula.

Let me try to define $ a_n = z^n + \frac{1}{z^n} $. Then, maybe I can find a recurrence relation for $ a_n $. Let me check for small n.

For $ n = 1 $, we have $ a_1 = z + 1/z = \frac{1 + \sqrt{5}}{2} $, which is given.

What about $ a_2 $? Let's compute $ a_2 = z^2 + 1/z^2 $. There's a formula for that in terms of $ a_1 $:

$ a_2 = (z + 1/z)^2 - 2 = a_1^2 - 2 $.

Similarly, for $ a_3 $, maybe there's a similar relation. Let me think. If I multiply $ a_1 $ and $ a_2 $, perhaps?

Wait, $ a_1 \times a_2 = (z + 1/z)(z^2 + 1/z^2) = z^3 + z^{-1} + z + z^{-3} $. That is $ a_3 + a_1 $. So $ a_1 a_2 = a_3 + a_1 $, which gives $ a_3 = a_1 a_2 - a_1 $.

Alternatively, there's a general recurrence relation for $ a_n $. Let me recall:

If $ a_{n+1} = a_1 a_n - a_{n-1} $.

Yes, that seems familiar. Let me verify this for n=2:

$ a_2 = a_1 a_1 - a_0 $. Wait, but what is $ a_0 $? Since $ a_0 = z^0 + 1/z^0 = 1 + 1 = 2 $.

So indeed, for n=1, $ a_1 = z + 1/z $, and for n=2, $ a_2 = a_1 a_1 - a_0 = (a_1)^2 - 2 $, which matches the earlier formula.

So the recurrence relation seems to be $ a_{n} = a_1 a_{n-1} - a_{n-2} $.

Therefore, if I can compute $ a_n $ up to n=85 using this recurrence, then I can find the desired value. However, computing up to 85 manually would be tedious. Maybe there's a pattern or periodicity?

Alternatively, perhaps the sequence $ a_n $ follows some periodic behavior? Let me check with the given value of $ a_1 $, which is the golden ratio, $ \phi = \frac{1 + \sqrt{5}}{2} $.

Let me compute the first few terms to see if a pattern emerges.

First, let me compute $ a_1 = \phi $.

Compute $ a_2 $:

$ a_2 = \phi^2 - 2 $.

But since $ \phi $ satisfies the equation $ \phi^2 = \phi + 1 $, so substituting:

$ a_2 = (\phi + 1) - 2 = \phi - 1 $.

Wait, $ \phi - 1 $ is actually $ \frac{1}{\phi} $. Because $ \phi = \frac{1 + \sqrt{5}}{2} $, then $ 1/\phi = \frac{\sqrt{5} - 1}{2} $, which is equal to $ \phi - 1 $. Let me confirm:

$ \phi - 1 = \frac{1 + \sqrt{5}}{2} - 1 = \frac{-1 + \sqrt{5}}{2} = \frac{\sqrt{5} - 1}{2} $, and $ 1/\phi = \frac{2}{1 + \sqrt{5}} \times \frac{\sqrt{5} - 1}{\sqrt{5} - 1} = \frac{2(\sqrt{5} - 1)}{5 - 1} = \frac{2(\sqrt{5} - 1)}{4} = \frac{\sqrt{5} - 1}{2} $. Yes, they are equal. Therefore, $ a_2 = 1/\phi $.

Now, moving on to $ a_3 $. Using the recurrence $ a_3 = a_1 a_2 - a_1 $. Wait, no, the recurrence is $ a_n = a_1 a_{n-1} - a_{n - 2} $. Therefore, for $ n = 3 $, $ a_3 = a_1 a_2 - a_1 $. Wait, $ a_1 a_2 - a_0 $? Wait, no. Let me check again.

Wait, general recurrence is $ a_n = a_1 a_{n-1} - a_{n-2} $. For n=2, it would be $ a_2 = a_1 a_1 - a_0 $, which is $ a_1^2 - 2 $. For n=3, it's $ a_3 = a_1 a_2 - a_1 $. Wait, but $ a_1 a_2 - a_1 $? Wait, that seems inconsistent. Let me check with the previous multiplication:

Earlier I had $ a_1 a_2 = a_3 + a_1 $, so rearranged, $ a_3 = a_1 a_2 - a_1 $. Therefore, for n=3, $ a_3 = a_1 a_2 - a_1 $. But according to the general recurrence, $ a_3 = a_1 a_2 - a_1 $, yes. So that works.

Therefore, $ a_3 = \phi \times (1/\phi) - \phi = 1 - \phi $. Wait, $ 1 - \phi $ is negative? Let me compute numerically:

$ \phi \approx 1.618 $, so $ 1 - \phi \approx -0.618 $, which is $ -1/\phi $. Let me verify:

Since $ \phi - 1 = 1/\phi \approx 0.618 $, so $ 1 - \phi = -1/\phi $. Therefore, $ a_3 = -1/\phi $.

Now let me compute $ a_4 $ using the recurrence $ a_4 = a_1 a_3 - a_2 $.

So substituting, $ a_4 = \phi \times (-1/\phi) - (1/\phi) = -1 - 1/\phi $. Wait, that seems messy. Let me check with another approach. Maybe using the identity $ a_4 = (z^2 + 1/z^2)^2 - 2 = a_2^2 - 2 $.

Let me compute that:

$ a_2 = 1/\phi $, so $ a_2^2 - 2 = (1/\phi)^2 - 2 $. But $ 1/\phi^2 = (\phi - 1)^2 = \phi^2 - 2\phi + 1 $. Wait, let me compute $ 1/\phi^2 $:

Since $ \phi^2 = \phi + 1 $, so $ 1/\phi^2 = 1/(\phi + 1) $. Alternatively, maybe another approach.

Alternatively, maybe I can use the recurrence relation. Let me proceed with the recurrence:

$ a_4 = a_1 a_3 - a_2 = \phi \times (-1/\phi) - (1/\phi) = -1 - 1/\phi $.

Wait, but maybe that's correct? Let me compute numerically:

$ \phi \approx 1.618 $, so $ -1 - 1/\phi \approx -1 - 0.618 = -1.618 $, which is -phi. Wait, so $ a_4 = -\phi $.

Alternatively, let me check using the other method: $ a_4 = a_2^2 - 2 $.

Since $ a_2 = 1/\phi \approx 0.618 $, then $ a_2^2 \approx 0.618^2 \approx 0.618 * 0.618. Let me compute:

$ 0.618^2 = (approx) 0.618 * 0.618. Let me recall that $ 1/\phi^2 = (\phi - 1)^2 = \phi^2 - 2\phi + 1. But since $ \phi^2 = \phi + 1, substitute: $ 1/\phi^2 = (\phi + 1) - 2\phi + 1 = -\phi + 2. Therefore, $ a_2^2 = 1/\phi^2 = 2 - \phi $. Hence, $ a_4 = (2 - \phi) - 2 = -\phi $. Yes! So that matches the previous result.

So $ a_4 = -\phi $.

Now $ a_5 = a_1 a_4 - a_3 = \phi \times (-\phi) - (-1/\phi) = -\phi^2 + 1/\phi $.

But $ \phi^2 = \phi + 1 $, so substitute:

$ -(\phi + 1) + 1/\phi = -\phi -1 + 1/\phi $. But earlier we know that $ 1/\phi = \phi - 1 $. Therefore, substituting:

$ -\phi -1 + (\phi - 1) = (-\phi + \phi) + (-1 -1) = 0 - 2 = -2 $.

So $ a_5 = -2 $.

Interesting. Let's compute $ a_6 $ using the recurrence relation:

$ a_6 = a_1 a_5 - a_4 = \phi \times (-2) - (-\phi) = -2\phi + \phi = -\phi $.

Wait, so $ a_6 = -\phi $. But earlier, $ a_4 = -\phi $. Let me check with another method. Maybe $ a_6 = a_2 a_5 - a_4 $? Wait, no, the recurrence is $ a_n = a_1 a_{n-1} - a_{n-2} $. So for n=6: $ a_6 = a_1 a_5 - a_4 $.

Which is as above. So $ a_6 = -\phi $.

Now $ a_7 = a_1 a_6 - a_5 = \phi \times (-\phi) - (-2) = -\phi^2 + 2 $.

Again, $ \phi^2 = \phi + 1 $, so substituting: $ -(\phi + 1) + 2 = -\phi -1 + 2 = -\phi + 1 $.

Which is $ 1 - \phi = -1/\phi $. Therefore, $ a_7 = -1/\phi $.

Wait, earlier $ a_3 = -1/\phi $, so $ a_7 = a_3 $. Interesting.

Then $ a_8 = a_1 a_7 - a_6 = \phi \times (-1/\phi) - (-\phi) = -1 + \phi $.

Which is $ \phi - 1 = 1/\phi $. So $ a_8 = 1/\phi $, which is $ a_2 $.

Similarly, $ a_9 = a_1 a_8 - a_7 = \phi \times (1/\phi) - (-1/\phi) = 1 + 1/\phi $.

Which is $ 1 + 1/\phi $. But since $ 1 + 1/\phi = 1 + (\phi - 1) = \phi $. Therefore, $ a_9 = \phi $, which is $ a_1 $.

Wait a minute! So let me list the terms I have so far:

n:  0    1       2         3          4         5         6         7         8         9
a_n: 2  φ    1/φ    -1/φ     -φ        -2       -φ      -1/φ     1/φ      φ

So, looking at n=1: φ, n=9: φ. Then n=2: 1/φ, n=8: 1/φ. Then n=3: -1/φ, n=7: -1/φ. n=4: -φ, n=6: -φ. n=5: -2.

So seems like the sequence is periodic with period 8? Wait, from n=1 to n=9, which is 8 steps, and it's repeating? Let me check:

If a_1 = φ, then a_9 = φ. Then, if the sequence repeats every 8 terms, then a_1 = a_9 = a_17 = ... So, to compute a_85, perhaps we can find 85 modulo 8? Wait, but need to check the period.

Wait, but let's check n=9 is a_9 = φ = a_1, n=10 would be a_10 = a_1 * a_9 - a_8 = φ*φ - 1/φ. Let me compute that.

But φ^2 = φ + 1, so a_10 = φ + 1 - 1/φ. Since 1/φ = φ -1, then substituting:

a_10 = φ + 1 - (φ -1) = φ + 1 - φ + 1 = 2. Which is a_0. Then a_10 = 2, which is a_0.

Then a_11 = a_1 a_10 - a_9 = φ * 2 - φ = 2φ - φ = φ, which is a_1. Therefore, the sequence is periodic with period 10? Wait, but from a_0 to a_10, we have a_0 = 2, then a_1 = φ, ..., a_10 = 2. So the period is 10? But earlier from a_1 to a_9 is φ, and then a_10 = 2. Wait, but if the period is 10, then the sequence would repeat every 10 terms. But let me check:

If a_1 = a_11, and a_2 = a_12, etc. But earlier, a_10 = 2 = a_0. So the sequence {a_n} has period 10? Let me check n=10: 2, which is a_0. Then n=11: φ, which is a_1. So yes, the sequence is periodic with period 10 starting from n=0? So the period is 10.

But then if that's the case, then the sequence for n ≥ 0 is:

n:  0: 2

n:  1: φ

n:  2: 1/φ

n:  3: -1/φ

n:  4: -φ

n:  5: -2

n:  6: -φ

n:  7: -1/φ

n:  8: 1/φ

n:  9: φ

n: 10: 2

So yes, the cycle is 10. Therefore, the period is 10. Therefore, if I can compute n modulo 10, then I can find the value of a_n.

But wait, let me confirm with n=5: a_5 = -2, n=15 would be same as n=5? Let me compute a_15. But maybe it's faster to check the period.

Alternatively, since the recurrence relation is linear and homogeneous, the sequence is periodic if the characteristic equation has roots that are roots of unity or something similar. But given that we have this periodicity of 10, perhaps.

But in any case, given that the sequence has period 10, starting from n=0. So to find a_85, since 85 divided by 10 is 8 with remainder 5, because 10*8=80, 85-80=5. Therefore, a_85 = a_5 = -2.

Wait, so according to this, the answer is -2. But let me check again with the earlier terms.

Wait, let me check with n=5: a_5=-2, then n=15 would be a_15=a_5=-2, and so on. Therefore, since 85=10*8 +5, so a_85=a_5=-2. But let me check with another approach to confirm.

Alternatively, maybe there's a mistake in my period? Let me check for n=10: a_10=2, which is a_0. Then a_11=a_1. So the period is 10. So the sequence repeats every 10 terms. Therefore, 85 divided by 10 is 8 with remainder 5, so a_85 = a_5 = -2.

But wait, let me check for n=5 and n=15. Let me compute a_10 through recurrence:

We had up to n=9: a_9 = φ, then a_10 = a_1 * a_9 - a_8 = φ*φ - 1/φ = φ^2 - 1/φ. Since φ^2 = φ + 1, this becomes φ +1 - 1/φ. Since 1/φ = φ -1, substituting:

φ +1 - (φ -1) = φ +1 - φ +1 = 2. So a_10 = 2.

Then a_11 = a_1 * a_10 - a_9 = φ * 2 - φ = 2φ - φ = φ. Which is a_1.

Similarly, a_12 = a_1 * a_11 - a_10 = φ * φ - 2 = φ^2 - 2 = (φ +1) - 2 = φ -1 = 1/φ = a_2.

Therefore, the period is indeed 10. Therefore, the sequence repeats every 10 terms. Therefore, since 85 mod 10 is 5, then a_85 = a_5 = -2.

But let me check another term to be sure. Let's compute a_15. Since 15 mod 10 = 5, so a_15 = a_5 = -2. Let me compute a_15 via recurrence:

We can compute up to a_10=2, a_11=φ, a_12=1/φ, a_13= -1/φ, a_14=-φ, a_15=-2. Yes! So that's correct. Therefore, the pattern holds.

Therefore, answer is -2.

But let me check again if the initial assumption is correct. The problem says z is a complex number such that z + 1/z equals phi. But does that uniquely determine z? Or are there multiple z's? However, regardless of the value of z, the expression z^n + 1/z^n is determined by the recurrence relation, so even though there may be multiple z's, the value of a_n would be same.

Alternatively, if z is on the unit circle, then z = e^{iθ}, and z + 1/z = 2 cosθ. In this case, 2 cosθ = phi. But phi is approx 1.618, which is greater than 2? Wait, no! Wait 2 cosθ can't be more than 2. Wait, phi is approx 1.618, which is less than 2, so that's okay.

Wait, cosθ = phi / 2 ≈ 0.809, which is valid because cos(36°) ≈ 0.809, which is correct. Because phi = 2 cos(36°). So θ = 36° or θ = -36°, etc. Therefore, z could be e^{iπ/5} or e^{-iπ/5}, or other multiples.

But regardless, the expression z^n + 1/z^n would be 2 cos(nθ). Therefore, since θ = π/5 or -π/5, so for example, if θ = π/5, then a_n = 2 cos(nπ/5).

Therefore, let's check for n=1: 2 cos(π/5) = 2*(sqrt(5)+1)/4 = (sqrt(5)+1)/2 = phi. Correct.

Similarly, n=5: 2 cos(5π/5) = 2 cos(π) = -2. Which matches our earlier result of a_5=-2.

Similarly, n=85: 2 cos(85π/5) = 2 cos(17π). Since cos(17π) = cos(π + 16π) = cos(π) = -1. Therefore, 2*(-1) = -2. Which matches. Therefore, regardless of the approach, the answer is -2.

Therefore, even if I consider θ = π/5, then 85θ = 85π/5 = 17π, which is equivalent to π modulo 2π (since 17π = 8*2π + π). Therefore, cos(17π) = cos(π) = -1, so 2 cos(17π) = -2.

Therefore, answer is -2.

Hence, I can confirm that the answer is \boxed{-2}.

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

We are given the equation:

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

Let us define a sequence $\{a_n\}$ as follows:

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

From this definition, we can derive a recurrence relation for $a_n$ based on the identity:

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

This is a second-order linear recurrence relation, and it is satisfied by any number $z$ for which $z + 1/z = a_1$, as long as $z \ne 0$.

---

### Step 1: Determine the Initial Terms

We are given:

- $a_1 = \phi = \frac{1 + \sqrt{5}}{2}$
- $a_0 = z^0 + \frac{1}{z^0} = 1 + 1 = 2$

Using the recurrence, we compute:

- $a_2 = a_1^2 - 2 = \phi^2 - 2 = (\phi + 1) - 2 = \phi - 1 = \frac{1}{\phi}$
- $a_3 = a_1 a_2 - a_1 = \phi \cdot \frac{1}{\phi} - \phi = 1 - \phi = -\frac{1}{\phi}$
- $a_4 = a_1 a_3 - a_2 = \phi \cdot \left(-\frac{1}{\phi}\right) - \frac{1}{\phi} = -1 - \frac{1}{\phi} = -\phi$
- $a_5 = a_1 a_4 - a_3 = \phi \cdot (-\phi) - \left(-\frac{1}{\phi}\right) = -\phi^2 + \frac{1}{\phi} = -(\phi + 1) + \phi - 1 = -2$

---

### Step 2: Identify the Periodicity

Continuing the computation using the recurrence, we eventually find that the sequence $\{a_n\}$ is **periodic with period 10**, as follows:

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

So the sequence repeats every 10 terms:

$$
a_{n+10} = a_n
$$

---

### Step 3: Compute $a_{85}$

We compute the remainder of $85 \mod 10$:

$$
85 \div 10 = 8 \text{ remainder } 5 \Rightarrow 85 \equiv 5 \pmod{10}
$$

Therefore:

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

---

### Final Answer

$$
\boxed{-2}
$$

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