OurBigBook.comSite Source code

Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 6E / i / Solution

For

n = 1

,

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

If

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} .

Solved by gpt-5.6-sol high.

 Ancestors (11)