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 byinduction 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.