Talk:Cassini and Catalan identities

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

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

Recalling the formula for the Fibonacci numbers we know that

$$F_n = F_{n-1} + F_{n-2}$$

Therefore for $$n = m + 1$$

And for $$n = m + 2$$

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

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