Talk:Cassini and Catalan identities

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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}
= − Fm + 1Fm + 1 + FmFm + 2

Recalling the formula for the Fibonacci numbers we know that

Fn = Fn − 1 + Fn − 2

Therefore for n = m + 1

Fm + 1 = Fm + 1 − 1 + Fm + 1 − 2
= Fm + Fm − 1

And for n = m + 2

Fm + 2 = Fm + 2 − 1 + Fm + 2 − 2
= Fm + 1 + Fm

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

− Fm + 1Fm + 1 + FmFm + 2 = − Fm + 1(Fm + Fm − 1) + Fm(Fm + 1 + Fm)
= -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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