Prove Cassini’s identity?by induction

Question

Prove Cassini’s identity: fn1 · fn+1 − (fn)² = (−1)ⁿ for any n ≥ 2, there fn is the n-th Fibonacci number.

Answer 1

Cassini's identity states that for the Fibonicci sequence (F_n)_n≥1, F₁=F₂=1, there is always the equality F_n-1F_n+1−F_n² = (−1)ⁿ for n≥2, Since this is obviously true for n=2 as 1 x 2 − 1² = 1, and for n = 3 as 1 x 3 − 2² = −1, It suffices to show that

F_n-1F_n+1−F_n² = −(F_nF_n-2 − F_n-1²)_.

Now F_n-1F_n+1 − F_n² = F_n-1(F_n + F_n-1) − F_n(F_n-1 + F_n-2) = F_n-1F_n + F_n-1² − F_nF_n-1 − F_nF_n-2

= F_n-1² − F_nF_n-2 = − F_nF_n-2 + F_n-1² = −(F_nF_n-2 − F_n-1²)_.