 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ia/paper-4/6e/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

For

n = 1

F_{2} F_{0} - F_{1}^{2} = - 1 = (- 1)^{1} .

C_{n} = F_{n + 1} F_{n - 1} - F_{n}^{2}

, then the recurrence gives

C_{n + 1} = F_{n + 2} F_{n} - F_{n + 1}^{2} = (F_{n + 1} + F_{n}) F_{n} - F_{n + 1}^{2} = F_{n}^{2} - F_{n + 1} F_{n - 1} = - C_{n} .

Induction yields the Fibonacci determinant identity known as Cassini's identity:

F_{n + 1} F_{n - 1} - F_{n}^{2} = (- 1)^{n} .

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