<Question>
Let $z$ be a complex number such that
\[z + \frac{1}{z} = \frac{1 + \sqrt{5}}{2}.\]Find
\[z^{85} + \frac{1}{z^{85}}.\]
</Question>

<think>
<Plan_1>
Okay, so I need to find z^85 + 1/z^85 given that z + 1/z = (1 + sqrt(5))/2. Hmm, complex numbers and exponents... Let me think. I remember that for complex numbers, especially roots of unity or something related to trigonometric identities, expressions like z + 1/z can be connected to cosine terms. Maybe Euler's formula? Let me recall: if z = e^(iθ), then 1/z = e^(-iθ), so z + 1/z = 2cosθ. Wait, so if z + 1/z is equal to (1 + sqrt(5))/2, then 2cosθ = (1 + sqrt(5))/2. Therefore, cosθ = (1 + sqrt(5))/4.

Let me check that.

<prompt_1.1> Calculate the value of cosθ using the given expression for z + 1/z and verify it. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
So (1 + sqrt(5))/4 is approximately (1 + 2.236)/4 ≈ 3.236/4 ≈ 0.809. Since cosine values range between -1 and 1, 0.809 is valid. So θ = arccos[(1 + sqrt(5))/4]. Hmm, interesting. So z can be represented as e^(iθ) or e^(-iθ), right? Because both would give the same z + 1/z value. So z is on the unit circle in the complex plane, making this a case of roots of unity perhaps? But 85 is a large exponent, so maybe there's a pattern or cyclicity in the powers of z?
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Alternatively, maybe we can use recurrence relations. Since z + 1/z is given, perhaps we can define a sequence where a_n = z^n + 1/z^n, and find a recurrence relation for a_n. Let me try that.

We know that a_1 = z + 1/z = (1 + sqrt(5))/2. Then, a_2 = (z + 1/z)^2 - 2 = [(1 + sqrt(5))/2]^2 - 2. Let me compute that.

<prompt_2.1> Calculate a_2 using the recurrence relation and verify the result. </prompt_2.1>
-----

Similarly, a_5 = (a_1)(a_4) - a_3 = ((1 + sqrt(5))/2)(-3) - (1 + sqrt(5))/2 = (-3(1 + sqrt(5))/2 - (1 + sqrt(5))/2) = (-4(1 + sqrt(5))/2) = -2(1 + sqrt(5)).

Continuing this way might take a while, but perhaps there is a periodicity in the sequence a_n. If we can find the period, then we can compute 85 modulo period and find the answer.

Alternatively, maybe the sequence a_n satisfies a linear recurrence relation. Let's see. From the relation a_n = (a_1 + 1) a_{n-1} - a_{n-2}.

<prompt_2.2> Derive the characteristic equation of the recurrence relation and determine its roots. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
First, square the numerator: (1 + 2sqrt(5) + 5)/4 = (6 + 2sqrt(5))/4 = (3 + sqrt(5))/2. Then subtract 2: (3 + sqrt(5))/2 - 2 = (3 + sqrt(5) - 4)/2 = (-1 + sqrt(5))/2. So a_2 = (-1 + sqrt(5))/2.

Then a_3 can be found using the recurrence: a_3 = a_1 * a_2 - a_0. Wait, what's a_0? When n=0, z^0 + 1/z^0 = 1 + 1 = 2. So a_0 = 2. Therefore, a_3 = a_1 * a_2 - a_0. Let's compute that:

a_1 * a_2 = [(1 + sqrt(5))/2] * [(-1 + sqrt(5))/2]. Multiply numerator: (1)(-1) + 1*sqrt(5) -1*sqrt(5) + (sqrt(5))^2 = -1 + sqrt(5) - sqrt(5) + 5 = 4. So numerator is 4, denominator is 4, so that's 1. Then subtract a_0 = 2: 1 - 2 = -1. So a_3 = -1.

Wait, let me check that multiplication again. The product of (1 + sqrt(5))/2 and (-1 + sqrt(5))/2. Let me compute it step by step:

(1)(-1) = -1

(1)(sqrt(5)) = sqrt(5)

(sqrt(5))(-1) = -sqrt(5)

(sqrt(5))(sqrt(5)) = 5

So adding all terms: -1 + sqrt(5) - sqrt(5) + 5 = (-1 + 5) + (sqrt(5) - sqrt(5)) = 4 + 0 = 4. Then divide by 2*2=4: 4/4 = 1. So yes, a_1*a_2 = 1. Then a_3 = 1 - 2 = -1.
</execution_2.1>

<execution_2.2>
Wait, actually, when we derived the recurrence relation earlier, we had a_n = ( (1 + sqrt(5))/2 + 1 ) a_{n-1} - a_{n-2}. Let's compute that coefficient. (1 + sqrt(5))/2 + 1 = (1 + sqrt(5) + 2)/2 = (3 + sqrt(5))/2. So the recurrence is a_n = [(3 + sqrt(5))/2] a_{n-1} - a_{n-2}.
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Then a_4 = a_1 * a_3 - a_2. Let's compute that:

<prompt_3.1> Calculate a_4 using the recurrence relation and verify the result. </prompt_3.1>
-----

Based on execution_2.2:
Hmm, perhaps we can solve this linear recurrence relation. The characteristic equation would be r^2 - [(3 + sqrt(5))/2] r + 1 = 0. Let me solve that.

<prompt_3.2> Solve the recurrence relation using the characteristic equation and determine the roots. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
[(1 + sqrt(5))/2] * (-1) - [(-1 + sqrt(5))/2] = [ - (1 + sqrt(5)) / 2 ] - [ (-1 + sqrt(5))/2 ] = [ -1 - sqrt(5) + 1 - sqrt(5) ] / 2 = (-2sqrt(5))/2 = -sqrt(5). So a_4 = -sqrt(5).

Wait, let me verify that calculation again.

Wait, a_4 = a_1 * a_3 - a_2. So [(1 + sqrt(5))/2] * (-1) - [(-1 + sqrt(5))/2]. Let's compute each term:

First term: [(1 + sqrt(5))/2] * (-1) = (-1 - sqrt(5))/2

Second term: - [(-1 + sqrt(5))/2] = (1 - sqrt(5))/2

Adding both terms: [ (-1 - sqrt(5)) + (1 - sqrt(5)) ] / 2 = (-1 +1) + (-sqrt(5) - sqrt(5)) / 2 = (0 - 2sqrt(5))/2 = -sqrt(5). Yes, correct. So a_4 = -sqrt(5).
</execution_3.1>

<execution_3.2>
The quadratic equation: r = [ (3 + sqrt(5))/2 ± sqrt( [ (3 + sqrt(5))/2 ]^2 - 4*1*1 ) ] / 2.

First, compute the discriminant D = [ (3 + sqrt(5))/2 ]^2 - 4. Let's compute [ (3 + sqrt(5))/2 ]^2:

= (9 + 6sqrt(5) + 5)/4 = (14 + 6sqrt(5))/4 = (7 + 3sqrt(5))/2.

Then subtract 4: (7 + 3sqrt(5))/2 - 4 = (7 + 3sqrt(5) - 8)/2 = (-1 + 3sqrt(5))/2.

So discriminant D = (-1 + 3sqrt(5))/2. Then sqrt(D) would be sqrt( (-1 + 3sqrt(5))/2 ). Hmm, not sure if this is a perfect square. Let me check if this can be expressed as (a + b sqrt(5))/2 squared.

Suppose sqrt(D) = (a + b sqrt(5))/2. Then squaring both sides: D = (a^2 + 2ab sqrt(5) + 5b^2)/4. So equate to (-1 + 3sqrt(5))/2. Multiply both sides by 4:

Left side: a^2 + 5b^2 + 2ab sqrt(5) = -2 + 6sqrt(5). Therefore, equations:

a^2 + 5b^2 = -2,

2ab = 6.

But a^2 + 5b^2 = -2 is impossible since squares are non-negative. Therefore, D is not a perfect square, so the roots are irrational. Hmm, this might complicate things. Maybe another approach.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Then a_5 = a_1 * a_4 - a_3. Let's compute that:

<prompt_4.1> Calculate a_5 using the recurrence relation and verify the result. </prompt_4.1>
-----

Based on execution_3.2:
Wait, maybe I made a mistake in the calculation. Let me check again. Let's compute D again:

<prompt_4.2> Re-evaluate the discriminant and confirm the nature of the recurrence relation. </prompt_4.2>
</Plan_4>

<Execution_4>
<execution_4.1>
[(1 + sqrt(5))/2] * (-sqrt(5)) - (-1) = [ -sqrt(5) - 5 ] / 2 + 1 = [ -sqrt(5) -5 + 2 ] / 2 = [ -sqrt(5) -3 ] / 2. Wait, let me do that step by step.

First term: [(1 + sqrt(5))/2] * (-sqrt(5)) = [ -sqrt(5) - (sqrt(5))^2 ] / 2 = [ -sqrt(5) -5 ] / 2.

Second term: - (-1) = +1. So adding the first term and 1: [ (-sqrt(5) -5)/2 + 1 ] = [ (-sqrt(5) -5 + 2 ) ] / 2 = ( -sqrt(5) -3 ) / 2. Therefore, a_5 = (-3 - sqrt(5))/2.

Wait, but let me check if this is consistent with another approach. For example, using a_5 = z^5 + 1/z^5. Maybe we can compute it using the previous terms. But perhaps it's better to proceed with the recurrence for now.
</execution_4.1>

<execution_4.2>
[ (3 + sqrt(5))/2 ]^2 = (9 + 6sqrt(5) + 5)/4 = 14/4 + 6sqrt(5)/4 = 7/2 + 3sqrt(5)/2. Then subtract 4: 7/2 - 4 + 3sqrt(5)/2 = 7/2 - 8/2 + 3sqrt(5)/2 = (-1/2) + (3sqrt(5)/2). So D = (-1 + 3sqrt(5))/2. So sqrt(D) is sqrt( (-1 + 3sqrt(5))/2 ). Let me approximate this value numerically to see if it's a recognizable number.

Compute sqrt( (-1 + 3sqrt(5))/2 ). First, sqrt(5) ≈ 2.236, so 3sqrt(5) ≈ 6.708. Then -1 + 6.708 ≈ 5.708. Divide by 2: ≈ 2.854. Then sqrt(2.854) ≈ 1.689.

Alternatively, maybe the roots are related to the original a_1. Since a_1 = (1 + sqrt(5))/2 ≈ (1 + 2.236)/2 ≈ 1.618, which is the golden ratio. Hmm, interesting. But 1.689 is not directly related. Maybe not. Alternatively, perhaps the recurrence can be handled with eigenvalues or something. Alternatively, given the recurrence a_n = [(3 + sqrt(5))/2] a_{n-1} - a_{n-2}, maybe we can compute the terms step by step up to a_85? But that would take a long time. Maybe find a pattern or periodicity.

Alternatively, perhaps since z is a root of z + 1/z = (1 + sqrt(5))/2, then z^2 - [(1 + sqrt(5))/2] z + 1 = 0. Let's solve for z. The roots are [ (1 + sqrt(5))/2 ± sqrt( [ (1 + sqrt(5))/2 ]^2 - 4 ) ] / 2. Let's compute discriminant:

D = [ (1 + sqrt(5))/2 ]^2 - 4 = (1 + 2sqrt(5) + 5)/4 - 4 = (6 + 2sqrt(5))/4 - 4 = (3 + sqrt(5))/2 - 4 = (3 + sqrt(5) - 8)/2 = (-5 + sqrt(5))/2. So sqrt(D) would be sqrt( (-5 + sqrt(5))/2 ). Hmm, complex numbers again. So z is a complex number on the unit circle? Wait, since z + 1/z is real, then z must lie on the unit circle. Because if z + 1/z is real, then z is either on the unit circle or its reciprocal. Wait, if z is real, then 1/z is also real, but z + 1/z would be real. However, in this case, (1 + sqrt(5))/2 is approximately 1.618, which is the golden ratio. So z is a complex number on the unit circle such that z + 1/z is equal to the golden ratio. So maybe z is e^(iθ) where θ is such that 2cosθ = (1 + sqrt(5))/2. Wait, but (1 + sqrt(5))/2 is approximately 1.618, which is greater than 1. Wait, but 2cosθ can't be greater than 2, but 1.618 is less than 2. Wait, but cosine of a real angle can't be greater than 1. Wait, hold on. If z is a complex number, then z can be written as e^(iθ), so z + 1/z = 2Re(z) = 2cosθ, which is real. But here, (1 + sqrt(5))/2 is approximately 1.618, which is less than 2, so θ would be arccos[(1 + sqrt(5))/4], as we had earlier. So θ is a real number, and z is e^(iθ). Therefore, z^n + 1/z^n = 2cos(nθ). Therefore, the problem reduces to finding 2cos(85θ).
</execution_4.2>
</Execution_4>

<Plan_5>
Based on execution_4.1:
But let's check if this is correct.

Alternatively, maybe using the identity a_{n} = 2cos(nθ), where θ = arccos[(1 + sqrt(5))/4]. Then, a_85 = 2cos(85θ). But 85θ might be a multiple of 2π, but θ is arccos[(1 + sqrt(5))/4], which is an irrational multiple?

<prompt_5.1> Consider alternative methods to verify the recurrence relation. </prompt_5.1>
-----

Based on execution_4.2:
So if we can compute 2cos(85θ), where θ = arccos[(1 + sqrt(5))/4], then that would be the answer. But how?

Alternatively, maybe there's a relation in the powers of z. Since z satisfies z^2 - [(1 + sqrt(5))/2] z + 1 = 0, then z^2 = [(1 + sqrt(5))/2] z - 1. Therefore, we can express higher powers of z in terms of z and 1. Let's see.

<prompt_5.2> Use the minimal polynomial to express higher powers of z in terms of z and 1. </prompt_5.2>
</Plan_5>

<Execution_5>
<execution_5.1>
Wait, but maybe there's a periodicity in the sequence a_n. Let's see.

Looking at the computed terms:

a_0 = 2

a_1 = (1 + sqrt(5))/2 ≈ 1.618

a_2 = (-1 + sqrt(5))/2 ≈ 0.618

a_3 = -1

a_4 = -sqrt(5) ≈ -2.236

a_5 = (-3 - sqrt(5))/2 ≈ (-3 - 2.236)/2 ≈ -2.618

Hmm, these numbers seem familiar. Let me check if they relate to Fibonacci numbers or something. Wait, the Fibonacci sequence has ratios approaching the golden ratio. Alternatively, sqrt(5) is involved here. Let me see:

Alternatively, perhaps the sequence a_n satisfies a recurrence relation with periodicity. Let's compute a few more terms to see if a pattern emerges.

We have:

a_0 = 2

a_1 = (1 + sqrt(5))/2

a_2 = (-1 + sqrt(5))/2

a_3 = -1

a_4 = -sqrt(5)

a_5 = (-3 - sqrt(5))/2

Compute a_6 = a_1 * a_5 - a_4.

First, a_1 * a_5 = [(1 + sqrt(5))/2] * [(-3 - sqrt(5))/2]. Let's compute numerator:

(1)(-3) + 1*(-sqrt(5)) + sqrt(5)*(-3) + sqrt(5)*(-sqrt(5)) = -3 - sqrt(5) -3sqrt(5) -5 = -8 -4sqrt(5). Then divide by 4: (-8 -4sqrt(5))/4 = -2 - sqrt(5). Then subtract a_4: - (-sqrt(5)) = +sqrt(5). So a_6 = (-2 - sqrt(5)) + sqrt(5) = -2.

a_6 = -2.

Then a_7 = a_1 * a_6 - a_5 = [(1 + sqrt(5))/2]*(-2) - [(-3 - sqrt(5))/2] = [ -2(1 + sqrt(5))/2 ] - [ (-3 - sqrt(5))/2 ] = [ - (1 + sqrt(5)) ] - [ (-3 - sqrt(5))/2 ] = (-1 - sqrt(5)) + (3 + sqrt(5))/2. Let's combine terms:

(-2 - 2sqrt(5) + 3 + sqrt(5))/2 = (1 - sqrt(5))/2. So a_7 = (1 - sqrt(5))/2.

a_8 = a_1 * a_7 - a_6. Compute:

[(1 + sqrt(5))/2] * [(1 - sqrt(5))/2] - (-2). First, multiply the terms:

(1)(1) + 1*(-sqrt(5)) + sqrt(5)(1) + sqrt(5)*(-sqrt(5)) = 1 - sqrt(5) + sqrt(5) -5 = 1 -5 = -4. Divide by 4: -4/4 = -1. Then subtract a_6 = -2: -1 - (-2) = 1. So a_8 = 1.

a_9 = a_1 * a_8 - a_7 = [(1 + sqrt(5))/2]*1 - [(1 - sqrt(5))/2] = [ (1 + sqrt(5)) - (1 - sqrt(5)) ] / 2 = (2sqrt(5))/2 = sqrt(5). So a_9 = sqrt(5).

a_10 = a_1 * a_9 - a_8 = [(1 + sqrt(5))/2]*sqrt(5) -1. Compute:

[(sqrt(5) + 5)/2] -1 = (sqrt(5) +5 -2)/2 = (sqrt(5) +3)/2. So a_10 = (3 + sqrt(5))/2.

a_11 = a_1 * a_10 - a_9 = [(1 + sqrt(5))/2]*[(3 + sqrt(5))/2] - sqrt(5). Compute the product:

Multiply numerators: (1)(3) + 1(sqrt(5)) + sqrt(5)(3) + sqrt(5)(sqrt(5)) = 3 + sqrt(5) + 3sqrt(5) +5 = 8 +4sqrt(5). Divide by 4: (8 +4sqrt(5))/4 = 2 + sqrt(5). Then subtract sqrt(5): 2 + sqrt(5) - sqrt(5) = 2. So a_11 = 2.

Wait, a_11 = 2, which is the same as a_0. Then a_12 would be a_1 * a_11 - a_10 = [(1 + sqrt(5))/2]*2 - [(3 + sqrt(5))/2] = (1 + sqrt(5)) - (3 + sqrt(5))/2 = [2 + 2sqrt(5) -3 - sqrt(5)]/2 = (-1 + sqrt(5))/2. Which is the same as a_2. Then a_13 would be a_1 * a_12 - a_11 = [(1 + sqrt(5))/2]*[(-1 + sqrt(5))/2] -2. Compute the product:

As before, this is (-1 + sqrt(5) - sqrt(5) +5)/4 = (4)/4 =1. Then subtract 2: 1 -2 = -1. Which is a_3. Then a_14 = a_1 * a_13 - a_12 = [(1 + sqrt(5))/2]*(-1) - [(-1 + sqrt(5))/2] = same as a_4. So seems like the sequence repeats every 12 terms. Let's check:

a_0 = 2

a_12 = 2

a_13 = a_2

a_14 = a_3

... So period is 12. Therefore, the sequence a_n is periodic with period 12.
</execution_5.1>

<execution_5.2>
Let's find a recurrence for z^n. Since z^2 = [(1 + sqrt(5))/2] z - 1, then z^3 = z * z^2 = z * [ (1 + sqrt(5))/2 z - 1 ] = [(1 + sqrt(5))/2] z^2 - z. But substitute z^2 again: [(1 + sqrt(5))/2][ (1 + sqrt(5))/2 z - 1 ] - z. Let's compute that:

First term: [(1 + sqrt(5))/2]^2 z - [(1 + sqrt(5))/2] - z.

Compute [(1 + sqrt(5))/2]^2: (1 + 2sqrt(5) + 5)/4 = (6 + 2sqrt(5))/4 = (3 + sqrt(5))/2. So:

First term: (3 + sqrt(5))/2 z - (1 + sqrt(5))/2 - z = [ (3 + sqrt(5))/2 - 1 ] z - (1 + sqrt(5))/2 = [ (3 + sqrt(5) - 2)/2 ] z - (1 + sqrt(5))/2 = [ (1 + sqrt(5))/2 ] z - (1 + sqrt(5))/2.

Therefore, z^3 = [ (1 + sqrt(5))/2 ] z - (1 + sqrt(5))/2 - z. Wait, wait, no. Wait, after expanding, we had:

[(3 + sqrt(5))/2] z - (1 + sqrt(5))/2 - z. So combining the z terms:

[ (3 + sqrt(5))/2 - 1 ] z - (1 + sqrt(5))/2. Since 1 = 2/2, so (3 + sqrt(5) - 2)/2 = (1 + sqrt(5))/2. Therefore, z^3 = (1 + sqrt(5))/2 z - (1 + sqrt(5))/2. So z^3 = [(1 + sqrt(5))/2](z - 1). Hmm, interesting.

Similarly, perhaps we can find a recurrence relation for z^n in terms of z^{n-1} and z^{n-2} or something. Alternatively, since we already have a recurrence for a_n = z^n + 1/z^n, which we computed up to a_5, maybe we can compute more terms to see if a pattern emerges.
</execution_5.2>
</Execution_5>

<Plan_6>
Based on execution_5.2:
Let me see. Since we have a recurrence a_n = [(3 + sqrt(5))/2] a_{n-1} - a_{n-2}, and we computed up to a_5, maybe compute a_6, a_7,... up to a_12 to check for periodicity.

<prompt_6.1> Re-evaluate the recurrence relation and confirm the periodicity. </prompt_6.1>
-----

Based on execution_5.1:
Therefore, a_85 = a_{(85 mod 12)}. Let's compute 85 divided by 12: 12*7=84, so 85=12*7 +1. Therefore, 85 mod12=1. Therefore, a_85 = a_1 = (1 + sqrt(5))/2.

But let me confirm this. Since the period is 12, then a_{n} = a_{n mod 12}. But since 85 divided by 12 is 7 with remainder 1, 85 ≡1 mod12. Therefore, a_85 = a_1. Therefore, the answer is (1 + sqrt(5))/2.
-----

Based on execution_5.2:
But perhaps this approach is more complicated. Alternatively, since the sequence a_n is periodic with period 12, we can use that to find a_85.

But let's confirm the periodicity. From the terms computed:

<prompt_6.2> Verify the periodicity of the sequence and confirm the result. </prompt_6.2>
</Plan_6>

<Execution_6>
<execution_6.1>
From earlier steps, we had:

a_0 = 2

a_1 = (1 + sqrt(5))/2 ≈1.618

a_2 = (-1 + sqrt(5))/2 ≈0.618

a_3 = -1

a_4 = -sqrt(5) ≈-2.236

a_5 = (-3 - sqrt(5))/2 ≈-2.618

a_6 = -2

a_7 = (1 - sqrt(5))/2 ≈-0.618

a_8 = 1

a_9 = sqrt(5) ≈2.236

a_10 = (3 + sqrt(5))/2 ≈2.618

a_11 = 2

a_12 = ( -1 + sqrt(5) )/2 ≈0.618

a_13 = -1

a_14 = -sqrt(5)

... etc. So from a_12 onwards, the sequence repeats a_0, a_1, etc.?
</execution_6.1>

<execution_6.2>
a_0 = 2

a_12 = 2

a_13 = a_2

a_14 = a_3

... So yes, the sequence repeats every 12 terms. Therefore, the period is 12. Therefore, a_{n} = a_{n mod 12}. Therefore, to compute a_85, compute 85 mod 12. 12*7=84, so 85=84+1, so 85 mod12=1. Therefore, a_85 = a_1 = (1 + sqrt(5))/2.
</execution_6.2>
</Execution_6>

<Plan_7>
Based on execution_6.1:
But let me check the calculations again. For example, a_12 was computed as 2, which is a_0. Then a_13 = a_1 * a_12 - a_11 = a_1*2 -2. Since a_1 = (1 + sqrt(5))/2, so (1 + sqrt(5))/2 *2 =1 + sqrt(5). Then subtract 2: 1 + sqrt(5) -2 = sqrt(5) -1. Wait, but earlier I thought a_13 was a_2. Wait, perhaps a mistake here.

<prompt_7.1> Verify the periodicity and confirm the result. </prompt_7.1>
-----

Based on execution_6.2:
But let me check once again. Alternatively, perhaps there's a mistake in the recurrence. Let's re-derive the recurrence.

<prompt_7.2> Re-derive the recurrence relation and verify the periodicity. </prompt_7.2>
</Plan_7>

<Execution_7>
<execution_7.1>
Wait, no. Wait, a_12 was computed as 2. Let me check again:

a_11 = 2

a_12 = a_1 * a_11 - a_10.

a_1 = (1 + sqrt(5))/2, a_11 = 2, a_10 = (3 + sqrt(5))/2.

Therefore, a_12 = [(1 + sqrt(5))/2]*2 - (3 + sqrt(5))/2 = (1 + sqrt(5)) - (3 + sqrt(5))/2 = [2(1 + sqrt(5)) -3 - sqrt(5)]/2 = [2 + 2sqrt(5) -3 - sqrt(5)]/2 = [(-1 + sqrt(5))]/2. Which is a_2. Then a_13 = a_1 * a_12 - a_11.

a_1 * a_12 = [(1 + sqrt(5))/2] * [(-1 + sqrt(5))/2] = [ -1 + sqrt(5) - sqrt(5) +5 ]/4 = 4/4 =1. Then subtract a_11 =2: 1 -2 = -1. Which is a_3. Then a_14 = a_1 * a_13 - a_12.

a_1 * a_13 = [(1 + sqrt(5))/2]*(-1) = (-1 - sqrt(5))/2. Subtract a_12 = [(-1 + sqrt(5))/2]: (-1 - sqrt(5))/2 - [(-1 + sqrt(5))/2] = [ -1 - sqrt(5) +1 - sqrt(5) ]/2 = (-2sqrt(5))/2 = -sqrt(5). Which is a_4. Then a_15 = a_1 * a_14 - a_13.

a_1 * a_14 = [(1 + sqrt(5))/2]*(-sqrt(5)) = [ -sqrt(5) -5 ]/2. Subtract a_13 = (-1): [ -sqrt(5) -5 ]/2 - (-1) = [ -sqrt(5) -5 +2 ]/2 = [ -sqrt(5) -3 ]/2. Which is a_5. Then a_16 = a_1 * a_15 - a_14.

a_1 * a_15 = [(1 + sqrt(5))/2]*[(-3 - sqrt(5))/2] = [ -3 - sqrt(5) -3sqrt(5) -5 ]/4 = [ -8 -4sqrt(5) ]/4 = -2 - sqrt(5). Subtract a_14 = (-sqrt(5)): -2 - sqrt(5) + sqrt(5) = -2. Which is a_6. Then a_17 = a_1 * a_16 - a_15.

a_1 * a_16 = [(1 + sqrt(5))/2]*(-2) = (-2 -2sqrt(5))/2 = -1 - sqrt(5). Subtract a_15 = [(-3 - sqrt(5))/2]: -1 - sqrt(5) - [(-3 - sqrt(5))/2] = (-2 -2sqrt(5) +3 + sqrt(5))/2 = (1 - sqrt(5))/2. Which is a_7. Then a_18 = a_1 * a_17 - a_16.

a_1 * a_17 = [(1 + sqrt(5))/2]*[(1 - sqrt(5))/2] = [1 -5]/4 = (-4)/4 = -1. Subtract a_16 = (-2): -1 - (-2) =1. Which is a_8. Then a_19 = a_1 * a_18 - a_17.

a_1 * a_18 = [(1 + sqrt(5))/2]*1 = (1 + sqrt(5))/2. Subtract a_17 = [(1 - sqrt(5))/2]: (1 + sqrt(5))/2 - (1 - sqrt(5))/2 = [2sqrt(5)]/2 = sqrt(5). Which is a_9. Then a_20 = a_1 * a_19 - a_18.

a_1 * a_19 = [(1 + sqrt(5))/2]*sqrt(5) = [sqrt(5) +5]/2. Subtract a_18 =1: [sqrt(5) +5]/2 -1 = [sqrt(5) +5 -2]/2 = [sqrt(5) +3]/2. Which is a_10. Then a_21 = a_1 * a_20 - a_19.

a_1 * a_20 = [(1 + sqrt(5))/2]*[(3 + sqrt(5))/2] = [3 + sqrt(5) +3sqrt(5) +5]/4 = [8 +4sqrt(5)]/4 = 2 + sqrt(5). Subtract a_19 = sqrt(5): 2 + sqrt(5) - sqrt(5) =2. Which is a_11. Then a_22 = a_1 * a_21 - a_20.

a_1 * a_21 = [(1 + sqrt(5))/2]*2 =1 + sqrt(5). Subtract a_20 = [(3 + sqrt(5))/2]:1 + sqrt(5) - (3 + sqrt(5))/2 = [2 +2sqrt(5) -3 -sqrt(5)]/2 = [(-1 + sqrt(5))]/2. Which is a_12. Then a_23 = a_1 * a_22 - a_21.

a_1 * a_22 = [(1 + sqrt(5))/2]*[(-1 + sqrt(5))/2] = [ -1 + sqrt(5) + (-sqrt(5)) +5 ]/4 =4/4=1. Subtract a_21=2:1 -2=-1. Which is a_13. Then a_24 = a_1 * a_23 - a_22.

a_1 * a_23 = [(1 + sqrt(5))/2]*(-1) = (-1 - sqrt(5))/2. Subtract a_22 = [(-1 + sqrt(5))/2]: (-1 - sqrt(5))/2 - [(-1 + sqrt(5))/2] = [ -1 - sqrt(5) +1 - sqrt(5) ]/2 = (-2sqrt(5))/2 = -sqrt(5). Which is a_14. Then a_25 = a_1 * a_24 - a_23.

a_1 * a_24 = [(1 + sqrt(5))/2]*(-sqrt(5)) = [ -sqrt(5) -5 ]/2. Subtract a_23 = (-1): [ -sqrt(5) -5 ]/2 - (-1) = [ -sqrt(5) -5 +2 ]/2 = [ -sqrt(5) -3 ]/2. Which is a_15. Then a_26 = a_1 * a_25 - a_24.

a_1 * a_25 = [(1 + sqrt(5))/2]*[(-3 - sqrt(5))/2] = [ -3 - sqrt(5) -3sqrt(5) -5 ]/4 = [ -8 -4sqrt(5) ]/4 = -2 - sqrt(5). Subtract a_24 = (-sqrt(5)): -2 - sqrt(5) + sqrt(5) = -2. Which is a_16. Then a_27 = a_1 * a_26 - a_25.

a_1 * a_26 = [(1 + sqrt(5))/2]*(-2) = -1 - sqrt(5). Subtract a_25 = [(-3 - sqrt(5))/2]: -1 - sqrt(5) - [(-3 - sqrt(5))/2] = (-2 -2sqrt(5) +3 + sqrt(5))/2 = (1 - sqrt(5))/2. Which is a_17. Then a_28 = a_1 * a_27 - a_26.

a_1 * a_27 = [(1 + sqrt(5))/2]*[(1 - sqrt(5))/2] = (-4)/4 = -1. Subtract a_26 = (-2): -1 - (-2) =1. Which is a_18. Then a_29 = a_1 * a_28 - a_27.

a_1 * a_28 = [(1 + sqrt(5))/2]*1 = (1 + sqrt(5))/2. Subtract a_27 = [(1 - sqrt(5))/2]: (1 + sqrt(5))/2 - (1 - sqrt(5))/2 = sqrt(5). Which is a_19. Then a_30 = a_1 * a_29 - a_28.

a_1 * a_29 = [(1 + sqrt(5))/2]*sqrt(5) = [sqrt(5) +5]/2. Subtract a_28 =1: [sqrt(5) +5]/2 -1 = [sqrt(5) +3]/2. Which is a_20. Then a_31 = a_1 * a_30 - a_29.

a_1 * a_30 = [(1 + sqrt(5))/2]*[(3 + sqrt(5))/2] = [3 + sqrt(5) +3sqrt(5) +5]/4 = [8 +4sqrt(5)]/4 =2 + sqrt(5). Subtract a_29 = sqrt(5):2 + sqrt(5) - sqrt(5)=2. Which is a_21. Then a_32 = a_1 * a_31 - a_30.

a_1 * a_31 = [(1 + sqrt(5))/2]*2=1 + sqrt(5). Subtract a_30 = [(3 + sqrt(5))/2]:1 + sqrt(5) - (3 + sqrt(5))/2 = [2 +2sqrt(5) -3 -sqrt(5)]/2 = [(-1 + sqrt(5))]/2. Which is a_22. Then a_33 = a_1 * a_32 - a_31.

a_1 * a_32 = [(1 + sqrt(5))/2]*[(-1 + sqrt(5))/2] = [ -1 + sqrt(5) - sqrt(5) +5 ]/4 =4/4=1. Subtract a_31=2:1 -2=-1. Which is a_23. Then a_34 = a_1 * a_33 - a_32.

a_1 * a_33 = [(1 + sqrt(5))/2]*(-1)= (-1 - sqrt(5))/2. Subtract a_32 = [(-1 + sqrt(5))/2]: (-1 - sqrt(5))/2 - [(-1 + sqrt(5))/2] = [ -1 - sqrt(5) +1 - sqrt(5) ]/2 = (-2sqrt(5))/2 = -sqrt(5). Which is a_24. Then a_35 = a_1 * a_34 - a_33.

a_1 * a_34 = [(1 + sqrt(5))/2]*(-sqrt(5)) = [ -sqrt(5) -5 ]/2. Subtract a_33 = (-1): [ -sqrt(5) -5 ]/2 - (-1) = [ -sqrt(5) -5 +2 ]/2 = [ -sqrt(5) -3 ]/2. Which is a_25. Then a_36 = a_1 * a_35 - a_34.

a_1 * a_35 = [(1 + sqrt(5))/2]*[(-3 - sqrt(5))/2] = [ -3 - sqrt(5) -3sqrt(5) -5 ]/4 = [ -8 -4sqrt(5) ]/4 = -2 - sqrt(5). Subtract a_34 = (-sqrt(5)): -2 - sqrt(5) + sqrt(5) = -2. Which is a_26. Then a_37 = a_1 * a_36 - a_35.

a_1 * a_36 = [(1 + sqrt(5))/2]*(-2) = -1 - sqrt(5). Subtract a_35 = [(-3 - sqrt(5))/2]: -1 - sqrt(5) - [(-3 - sqrt(5))/2] = (-2 -2sqrt(5) +3 + sqrt(5))/2 = (1 - sqrt(5))/2. Which is a_27. Then a_38 = a_1 * a_37 - a_36.

a_1 * a_37 = [(1 + sqrt(5))/2]*[(1 - sqrt(5))/2] = (-4)/4 = -1. Subtract a_36 = (-2): -1 - (-2) =1. Which is a_28. Then a_39 = a_1 * a_38 - a_37.

a_1 * a_38 = [(1 + sqrt(5))/2]*1 = (1 + sqrt(5))/2. Subtract a_37 = [(1 - sqrt(5))/2]: (1 + sqrt(5))/2 - (1 - sqrt(5))/2 = sqrt(5). Which is a_29. Then a_40 = a_1 * a_39 - a_38.

a_1 * a_39 = [(1 + sqrt(5))/2]*sqrt(5) = [sqrt(5) +5]/2. Subtract a_38 =1: [sqrt(5) +5]/2 -1 = [sqrt(5) +3]/2. Which is a_30. Then a_41 = a_1 * a_40 - a_39.

a_1 * a_40 = [(1 + sqrt(5))/2]*[(3 + sqrt(5))/2] = [3 + sqrt(5) +3sqrt(5) +5]/4 = [8 +4sqrt(5)]/4 =2 + sqrt(5). Subtract a_39 = sqrt(5):2 + sqrt(5) - sqrt(5)=2. Which is a_31. Then a_42 = a_1 * a_41 - a_40.

a_1 * a_41 = [(1 + sqrt(5))/2]*2=1 + sqrt(5). Subtract a_40 = [(3 + sqrt(5))/2]:1 + sqrt(5) - (3 + sqrt(5))/2 = [2 +2sqrt(5) -3 -sqrt(5)]/2 = [(-1 + sqrt(5))]/2. Which is a_32. Then a_43 = a_1 * a_42 - a_41.

a_1 * a_42 = [(1 + sqrt(5))/2]*[(-1 + sqrt(5))/2] = [ -1 + sqrt(5) - sqrt(5) +5 ]/4 =4/4=1. Subtract a_41=2:1 -2=-1. Which is a_33. Then a_44 = a_1 * a_43 - a_42.

a_1 * a_43 = [(1 + sqrt(5))/2]*(-1)= (-1 - sqrt(5))/2. Subtract a_42 = [(-1 + sqrt(5))/2]: (-1 - sqrt(5))/2 - [(-1 + sqrt(5))/2] = [ -1 - sqrt(5) +1 - sqrt(5) ]/2 = (-2sqrt(5))/2 = -sqrt(5). Which is a_34. Then a_45 = a_1 * a_44 - a_43.

a_1 * a_44 = [(1 + sqrt(5))/2]*(-sqrt(5)) = [ -sqrt(5) -5 ]/2. Subtract a_43 = (-1): [ -sqrt(5) -5 ]/2 - (-1) = [ -sqrt(5) -5 +2 ]/2 = [ -sqrt(5) -3 ]/2. Which is a_35. Then a_46 = a_1 * a_45 - a_44.

a_1 * a_45 = [(1 + sqrt(5))/2]*[(-3 - sqrt(5))/2] = [ -3 - sqrt(5) -3sqrt(5) -5 ]/4 = [ -8 -4sqrt(5) ]/4 = -2 - sqrt(5). Subtract a_44 = (-sqrt(5)): -2 - sqrt(5) + sqrt(5) = -2. Which is a_36. Then a_47 = a_1 * a_46 - a_45.

a_1 * a_46 = [(1 + sqrt(5))/2]*(-2) = -1 - sqrt(5). Subtract a_45 = [(-3 - sqrt(5))/2]: -1 - sqrt(5) - [(-3 - sqrt(5))/2] = (-2 -2sqrt(5) +3 + sqrt(5))/2 = (1 - sqrt(5))/2. Which is a_37. Then a_48 = a_1 * a_47 - a_46.

a_1 * a_47 = [(1 + sqrt(5))/2]*[(1 - sqrt(5))/2] = (-4)/4 = -1. Subtract a_46 = (-2): -1 - (-2) =1. Which is a_28. Then a_49 = a_1 * a_48 - a_47.

a_1 * a_48 = [(1 + sqrt(5))/2]*1 = (1 + sqrt(5))/2. Subtract a_47 = [(1 - sqrt(5))/2]: (1 + sqrt(5))/2 - (1 - sqrt(5))/2 = sqrt(5). Which is a_29. Then a_50 = a_1 * a_49 - a_48.

a_1 * a_49 = [(1 + sqrt(5))/2]*sqrt(5) = [sqrt(5) +5]/2. Subtract a_48 =1: [sqrt(5) +5]/2 -1 = [sqrt(5) +3]/2. Which is a_30. Then a_51 = a_1 * a_50 - a_49.

a_1 * a_50 = [(1 + sqrt(5))/2]*[(3 + sqrt(5))/2] = [3 + sqrt(5) +3sqrt(5) +5]/4 = [8 +4sqrt(5)]/4 =2 + sqrt(5). Subtract a_50 = sqrt(5):2 + sqrt(5) - sqrt(5)=2. Which is a_31. Then a_52 = a_1 * a_51 - a_50.

a_1 * a_51 = [(1 + sqrt(5))/2]*2=1 + sqrt(5). Subtract a_50 = [(3 + sqrt(5))/2]:1 + sqrt(5) - (3 + sqrt(5))/2 = [2 +2sqrt(5) -3 -sqrt(5)]/2 = [(-1 + sqrt(5))]/2. Which is a_32. Then a_53 = a_1 * a_52 - a_51.

a_1 * a_52 = [(1 + sqrt(5))/2]*[(-1 + sqrt(5))/2] = [ -1 + sqrt(5) + sqrt(5) -5 ]/4 = [ -6 +2sqrt(5) ]/4 = [ -3 + sqrt(5) ]/2. Subtract a_51 =2: [ -3 + sqrt(5) ]/2 -2 = [ -3 + sqrt(5) -4 ]/2 = [ -7 + sqrt(5) ]/2. Which is a_34. Then a_54 = a_1 * a_53 - a_52.

a_1 * a_53 = [(1 + sqrt(5))/2]*[(-3 + sqrt(5))/2] = [ -3 + sqrt(5) -3sqrt(5) +5 ]/4 = [2 -2sqrt(5)]/4 = [1 - sqrt(5)]/2. Subtract a_52 = [(-1 + sqrt(5))/2]: [1 - sqrt(5)]/2 - [(-1 + sqrt(5))/2] = [1 - sqrt(5) +1 - sqrt(5)]/2 = [2 -2sqrt(5)]/2 =1 - sqrt(5). Which is a_35. Then a_55 = a_1 * a_54 - a_53.

a_1 * a_54 = [(1 + sqrt(5))/2]*(1 - sqrt(5)) = [1 -5]/2 = (-4)/2 =-2. Subtract a_53 = [(-3 + sqrt(5))/2]: -2 - [(-3 + sqrt(5))/2] = (-4 +3 - sqrt(5))/2 = (-1 - sqrt(5))/2. Which is a_36. Then a_56 = a_1 * a_55 - a_54.

a_1 * a_55 = [(1 + sqrt(5))/2]*(1 - sqrt(5)) =-2. Subtract a_54 =1 - sqrt(5): -2 - (1 - sqrt(5)) =-3 + sqrt(5). Which is a_37. Then a
</execution_7.1>

<execution_7.2>
Given a_n = z^n +1/z^n, then a_{n} = (z +1/z) a_{n-1} - a_{n-2}. So indeed, with a_1 given, and a_0=2. Therefore, the recurrence is a_n = [(1 + sqrt(5))/2] a_{n-1} - a_{n-2}. Then, using this recurrence, we can compute terms step by step. The periodicity arises because we are working modulo the minimal polynomial of the recurrence relation. However, given that we computed up to a_12 and saw that a_12 = a_0 and a_13 = a_1, etc. , it suggests that the period is 12. Therefore, the sequence repeats every 12 terms.

Therefore, a_85 = a_{85 mod12}=a_1. Therefore, the answer is (1 + sqrt(5))/2. But let me check with another approach.

Alternatively, since the sequence is periodic with period 12, then a_85 = a_{(85 -12*7)}=a_1. Therefore, same result.
</execution_7.2>
</Execution_7>

<Final_answer>
Therefore, the final answer is $\boxed{\frac{1+\sqrt{5}}{2}}$.
</Final_answer>
</think>