math lessons - Cassini's identity

Cassini's identity

Cassini's identity is a mathematical identity for the Fibonacci numbers. It is a special case of Catalan's identity , with $r = 1$ . The identity states that for the n-th Fibonacci number,

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

Proof 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 n-th power of a matrix with determinant −1.

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

$F_{m}F_{m+2} - F_{m+1}^2$	$= -F_{m+1}^2 + F_{m}F_{m+2}$
	$= - F m + 1 F m + 1 + F m F m + 2$

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

01-04-2007 01:18:14
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Cassini's identity

Proof by matrix theory

Direct proof, by mathematical induction

Mathematics Encyclopedia and Lessons

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Cassini's identity

Proof by matrix theory

Direct proof, by mathematical induction

Popular
Subjects