Talk:Cassini and Catalan identities

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

Direct proof, by mathematical induction[edit]

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}