Problem
ALG-B1-M05-P014 A Small Cassini Identity
#14
★★★★☆ Level 4 of 5
For Fibonacci numbers \(F_1=F_2=1\), prove \(F_{n+1}F_{n-1}-F_n^2=(-1)^n\) for \(n\ge2\).
Check the base case and use \(F_{n+2}=F_{n+1}+F_n\) for the step.
Base \(n=2\): \(F_3F_1-F_2^2=2\cdot1-1=1=(-1)^2\).
If \(F_{n+1}F_{n-1}-F_n^2=(-1)^n\), then \(F_{n+2}F_n-F_{n+1}^2=(F_{n+1}+F_n)F_n-F_{n+1}^2=-(F_{n+1}F_{n-1}-F_n^2)=(-1)^{n+1}\).
Classic recurrence induction.