proof of Cassini’s identity

For all positive integers i, let Fi denote the ith Fibonacci numberMathworldPlanetmath, with F1=F2=1. We will show by inductionMathworldPlanetmath that the identity


holds for all positive integers n2. When n=2, we can substitute in the values for F1, F2 and F3 yielding the statement 2×1-12=(-1)2, which is true. Now suppose that the theoremMathworldPlanetmath is true when n=m, for some integer m2. Recalling the recurrence relation for the Fibonacci numbers, Fi+1=Fi+Fi-1, we have

Fm+2Fm-Fm+12 = (Fm+1+Fm)Fm-(Fm+Fm-1)2
= Fm+1Fm+Fm2-Fm2-2FmFm-1-Fm-12
= Fm+1Fm-2FmFm-1-Fm-12
= (Fm+Fm-1)Fm-2FmFm-1-Fm-12
= Fm2+Fm-1Fm-2FmFm-1-Fm-12
= Fm2-FmFm-1-Fm-12
= Fm2-(Fm+Fm-1)Fm-1
= Fm2-Fm+1Fm-1
= -(-1)m

by the induction hypothesis. So we get Fm+2Fm-Fm+12=(-1)m+1, and the result is thus true for n=m+1. The theorem now follows by induction.

Title proof of Cassini’s identity
Canonical name ProofOfCassinisIdentity
Date of creation 2013-03-22 14:44:40
Last modified on 2013-03-22 14:44:40
Owner yark (2760)
Last modified by yark (2760)
Numerical id 24
Author yark (2760)
Entry type Proof
Classification msc 11B39
Related topic CatalansIdentity