Cassini and Catalan identities

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Cassini's identity and Catalan's identity are mathematical identities for the Fibonacci numbers. The former is a special case of the latter, and states that for the nth Fibonacci number,

$F_{n-1}F_{n+1} - F_n^2 = (-1)^n.\,$

[edit] Proofs by matrix theory

A quick proof may be given by recognising the LHS as a determinant of a 2×2 matrix of Fibonacci numbers. The result is almost immediate when the matrix is seen to be the nth power of a matrix with determinant −1.

This proof generalizes immediately to give Catalan's identity:

$F_n^2 - F_{n-r}F_{n+r} = (-1)^{n+r}F_r^2.\,$