# Talk:Cassini and Catalan identities

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Field: Number theory

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

## Direct proof, by mathematical induction

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

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

Rewriting the equation for an easier understanding we have that

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

Recalling the formula for the Fibonacci numbers we know that

${\displaystyle F_{n}=F_{n-1}+F_{n-2}}$

Therefore for ${\displaystyle n=m+1}$

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

And for ${\displaystyle n=m+2}$

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

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

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