Talk:Cassini and Catalan identities

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

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

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

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