If x + 1/x = 3 (x ≠ 0), determine the exact value of x^5 + 1/x^5 using power-sum recurrences.

Introduction / Context:
When a relation like x + 1/x = k is given, higher symmetric power sums (x^n + 1/x^n) can be built efficiently via a recurrence. This avoids tedious expansion and keeps arithmetic clean and reliable.

Given Data / Assumptions:

• x + 1/x = 3 with x ≠ 0.
• We need x^5 + 1/x^5.

Concept / Approach:
Use the standard recurrence S_n = (x + 1/x) * S_{n−1} − S_{n−2}, where S_n = x^n + 1/x^n, S_0 = 2, and S_1 = x + 1/x. This recurrence is derived from multiplying by x + 1/x and regrouping terms.

Step-by-Step Solution:
S_0 = 2, S_1 = 3.S_2 = 3*S_1 − S_0 = 3*3 − 2 = 7.S_3 = 3*S_2 − S_1 = 3*7 − 3 = 18.S_4 = 3*S_3 − S_2 = 3*18 − 7 = 47.S_5 = 3*S_4 − S_3 = 3*47 − 18 = 141 − 18 = 123.

Verification / Alternative check:
One can also compute x^2 + 1/x^2 = 9 − 2 = 7, then use (x + 1/x)(x^4 + 1/x^4) − (x^3 + 1/x^3) patterns to reach the same number, confirming the recurrence path.

Why Other Options Are Wrong:

• 112, 102, 92, 83: These typically result from a single missed subtraction by S_{n−2} at one step in the recurrence.

Common Pitfalls:
Forgetting S_0 = 2; using S_n = (x + 1/x)*S_{n−1} only (without subtracting S_{n−2}).

Final Answer:
123