Talk:Cassini and Catalan identities

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Please add useful comments here--Cronholm144 08:56, 24 May 2007 (UTC)

Warehousing this proof. Charles Matthews 11:58, 25 October 2005 (UTC)

[edit] Direct proof, by mathematical induction

For $n = m + 1$ the result must be $( - 1) m + 1$ . Replacing in the equation we have

$F_{m+1-1}F_{m+1+1} - F_{m+1}^2 = F_{m}F_{m+2} - F_{m+1}^2$

Rewriting the equation for an easier understanding we have that

$F_{m}F_{m+2} - F_{m+1}^2$	$= -F_{m+1}^2 + F_{m}F_{m+2}$
	$= - F m + 1 F m + 1 + F m F m + 2$

Recalling the formula for the Fibonacci numbers we know that

$F n = F n - 1 + F n - 2$

Therefore for $n = m + 1$

$F m + 1$	$= F m + 1 - 1 + F m + 1 - 2$
	$= F m + F m - 1$

And for $n = m + 2$

$F m + 2$	$= F m + 2 - 1 + F m + 2 - 2$
	$= F m + 1 + F m$

Replacing these two known values in the equation we now have that