proof of Cassini’s identity

For all positive integers $i$ , let $F_{i}$ denote the $i^{th}$ Fibonacci number, with $F_{1}=F_{2}=1$ . We will show by induction that the identity

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

holds for all positive integers $n\geq 2$ . When $n=2$ , we can substitute in the values for $F_{1}$ , $F_{2}$ and $F_{3}$ yielding the statement $2\times 1-1^{2}=(-1)^{2}$ , which is true. Now suppose that the theorem is true when $n=m$ , for some integer $m\geq 2$ . Recalling the recurrence relation for the Fibonacci numbers, $F_{i+1}=F_{i}+F_{i-1}$ , we have

$\displaystyle F_{m+2}F_{m}-F_{m+1}^{2}$	$\displaystyle=$	$\displaystyle(F_{m+1}+F_{m})F_{m}-(F_{m}+F_{m-1})^{2}$
	$\displaystyle=$	$\displaystyle F_{m+1}F_{m}+F_{m}^{2}-F_{m}^{2}-2F_{m}F_{m-1}-F_{m-1}^{2}$
	$\displaystyle=$	$\displaystyle F_{m+1}F_{m}-2F_{m}F_{m-1}-F_{m-1}^{2}$
	$\displaystyle=$	$\displaystyle(F_{m}+F_{m-1})F_{m}-2F_{m}F_{m-1}-F_{m-1}^{2}$
	$\displaystyle=$	$\displaystyle F_{m}^{2}+F_{m-1}F_{m}-2F_{m}F_{m-1}-F_{m-1}^{2}$
	$\displaystyle=$	$\displaystyle F_{m}^{2}-F_{m}F_{m-1}-F_{m-1}^{2}$
	$\displaystyle=$	$\displaystyle F_{m}^{2}-(F_{m}+F_{m-1})F_{m-1}$
	$\displaystyle=$	$\displaystyle F_{m}^{2}-F_{m+1}F_{m-1}$
	$\displaystyle=$	$\displaystyle-(-1)^{m}$

by the induction hypothesis. So we get $F_{m+2}F_{m}-F_{m+1}^{2}=(-1)^{m+1}$ , 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