Okay, let's see. I need to find the value of (a₉)⁹ given that the sequence is defined by a_{i+1} = 1/(1 - a_i) and a₃ = a₁. Hmm, so the sequence is periodic with period 2? Wait, because if a₃ is equal to a₁, then maybe the sequence repeats every two terms? Let me check that.

First, let's try to write out the terms in terms of a₁. Let me denote a₁ as x for simplicity. Then:

a₂ = 1/(1 - x)

a₃ = 1/(1 - a₂) = 1/(1 - 1/(1 - x))

Let me simplify that. The denominator is 1 - 1/(1 - x). Let's compute that:

1 - 1/(1 - x) = ( (1 - x) - 1 ) / (1 - x ) = (-x)/(1 - x )

Therefore, a₃ = 1 / ( (-x)/(1 - x) ) = (1 - x)/(-x) = (x - 1)/x

But we are given that a₃ = a₁ = x. So:

(x - 1)/x = x

Multiply both sides by x:

x - 1 = x²

Bring all terms to one side:

x² - x + 1 = 0

Wait, that's a quadratic equation. Let's compute the discriminant:

D = (-1)² - 4*1*1 = 1 - 4 = -3

So the roots are complex numbers. Hmm, but the problem doesn't specify whether the sequence is real or complex. If it's complex, then maybe we can proceed, but if it's intended to be real, then this would be a problem. Wait, maybe I made a mistake in the algebra.

Let me check again:

a₃ = 1/(1 - a₂) = 1/(1 - 1/(1 - x))

Compute denominator first: 1 - 1/(1 - x) = ( (1 - x) - 1 ) / (1 - x ) = (-x)/(1 - x )

So 1 divided by that is (1 - x)/(-x) = (x - 1)/x. So that part is correct.

Set that equal to x:

(x - 1)/x = x

Multiply both sides by x:

x - 1 = x²

Which gives x² - x + 1 = 0. So indeed, complex roots. Hmm. Maybe the problem allows complex numbers? The problem didn't specify, but since it's asking for (a₉)^9, maybe the ninth power will result in a real number even if a₉ is complex. Let's proceed.

So the roots are x = [1 ± sqrt(-3)]/2 = (1 ± i√3)/2. These are complex numbers. So a₁ is one of these roots. Let's choose one, say (1 + i√3)/2. Then the sequence will have periodicity, but let's see.

Wait, if a₁ is (1 + i√3)/2, then a₂ is 1/(1 - a₁). Let's compute that:

1 - a₁ = 1 - (1 + i√3)/2 = (2 - 1 - i√3)/2 = (1 - i√3)/2

Therefore, a₂ = 1 / ( (1 - i√3)/2 ) = 2/(1 - i√3 )

Multiply numerator and denominator by (1 + i√3):

2*(1 + i√3)/( (1 - i√3)(1 + i√3) ) = 2*(1 + i√3)/(1 + 3) = 2*(1 + i√3)/4 = (1 + i√3)/2

Wait, so a₂ = (1 + i√3)/2, which is equal to a₁. Then a₃ = 1/(1 - a₂) = 1/(1 - (1 + i√3)/2) = same as before, which would be (1 - i√3)/2 divided by 1. Wait, but that's the same as a₂. Wait, no, let me compute that again.

Wait, if a₂ = (1 + i√3)/2, then 1 - a₂ = 1 - (1 + i√3)/2 = (2 - 1 - i√3)/2 = (1 - i√3)/2

Then a₃ = 1/(1 - a₂) = 1 / ( (1 - i√3)/2 ) = 2/(1 - i√3) = same as a₂, which is (1 + i√3)/2. Wait, so then a₃ = a₂, but we were given that a₃ = a₁. So this seems contradictory.

Wait, maybe I made a mistake here. Let's go back. If a₁ is (1 + i√3)/2, then a₂ is 2/(1 - i√3) which we found to be (1 + i√3)/2. So a₂ = a₁. Then a₃ = 1/(1 - a₂) = 1/(1 - a₁) = a₂. So then a₃ = a₂ = a₁. So in this case, the sequence would be constant? But then a₃ = a₁, which is given. But if the sequence is constant, then all terms are equal to a₁. So then a₉ would be a₁ as well, so (a₉)^9 = (a₁)^9.

But wait, is that the case? Let me check again.

If a₁ = (1 + i√3)/2, then a₂ = 1/(1 - a₁) = 1/(1 - (1 + i√3)/2 ) = 1/( (2 - 1 - i√3)/2 ) = 1/( (1 - i√3)/2 ) = 2/(1 - i√3 ) = multiply numerator and denominator by (1 + i√3):

2*(1 + i√3)/(1 + 3) = (2 + 2i√3)/4 = (1 + i√3)/2 = a₁. So a₂ = a₁. Then a₃ = 1/(1 - a₂) = 1/(1 - a₁) = a₂ = a₁. So indeed, the sequence is constant. Therefore, all terms are equal to a₁. Therefore, a₉ = a₁. So then (a₉)^9 = (a₁)^9.

But then, if a₁ is (1 + i√3)/2, then (a₁)^9. Let's compute that.

First, note that (1 + i√3)/2 is a complex number on the unit circle. Let's compute its modulus and argument.

Modulus: sqrt( (1/2)^2 + (√3/2)^2 ) = sqrt(1/4 + 3/4 ) = sqrt(1) = 1.

Argument: arctangent( (√3/2)/(1/2) ) = arctangent(√3) = π/3. So a₁ = e^{iπ/3}.

Therefore, (a₁)^9 = e^{iπ/3 *9} = e^{i3π} = cos(3π) + i sin(3π) = -1 + 0i = -1.

Similarly, if a₁ were (1 - i√3)/2, then it would be e^{-iπ/3}, and (a₁)^9 = e^{-i3π} = cos(-3π) + i sin(-3π) = cos(3π) - i sin(3π) = -1. So in either case, (a₁)^9 = -1.

Therefore, (a₉)^9 = -1.

Wait, so the answer is -1? Let me check again.

Alternatively, if the sequence is constant, all terms are equal to a₁. Then a₉ = a₁, so (a₉)^9 = (a₁)^9. Since a₁ is a root of x² - x + 1 = 0, then x³ = 1. Wait, is that true? Let's check:

Let x = (1 + i√3)/2. Then x³. Let's compute x³.

x = e^{iπ/3}, so x³ = e^{iπ} = -1. Similarly, x³ = -1. Therefore, (x)^9 = (x³)^3 = (-1)^3 = -1. So indeed, (a₉)^9 = -1.

Alternatively, since x² = x - 1 (from the equation x² - x + 1 = 0), then x³ = x * x² = x(x - 1) = x² - x = (x - 1) - x = -1. Therefore, x³ = -1, so x⁹ = (x³)^3 = (-1)^3 = -1. So either way, it's -1.

Therefore, the answer is -1.

Wait, but the problem didn't specify whether the terms are real or complex. If we assume real numbers, then the quadratic equation x² - x + 1 = 0 has no real roots, which would mean that there is no real sequence satisfying the given recurrence and a₃ = a₁. Therefore, the problem must be intended to have complex numbers. So the answer is -1.

Therefore, the final answer is \boxed{-1}.

**Final Answer**
\boxed{-1}
Therefore, the value of $(a_9)^9$ is $\boxed{-1}$.
</think>