proof of Cassini’s identity

For all positive integers $i$, let $F_{i}$ denote the $i^{th}$ Fibonacci number, with $F_{1}=F_{2}=1$. We will show by induction that the identity

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

holds for all positive integers $n\geq 2$. When $n=2$, we can substitute in the values for $F_{1}$, $F_{2}$ and $F_{3}$ yielding the statement $2\times 1-1^{2}=(-1)^{2}$, which is true. Now suppose that the theorem is true when $n=m$, for some integer $m\geq 2$. Recalling the recurrence relation for the Fibonacci numbers, $F_{i+1}=F_{i}+F_{i-1}$, we have

 $\displaystyle F_{m+2}F_{m}-F_{m+1}^{2}$ $\displaystyle=$ $\displaystyle(F_{m+1}+F_{m})F_{m}-(F_{m}+F_{m-1})^{2}$ $\displaystyle=$ $\displaystyle F_{m+1}F_{m}+F_{m}^{2}-F_{m}^{2}-2F_{m}F_{m-1}-F_{m-1}^{2}$ $\displaystyle=$ $\displaystyle F_{m+1}F_{m}-2F_{m}F_{m-1}-F_{m-1}^{2}$ $\displaystyle=$ $\displaystyle(F_{m}+F_{m-1})F_{m}-2F_{m}F_{m-1}-F_{m-1}^{2}$ $\displaystyle=$ $\displaystyle F_{m}^{2}+F_{m-1}F_{m}-2F_{m}F_{m-1}-F_{m-1}^{2}$ $\displaystyle=$ $\displaystyle F_{m}^{2}-F_{m}F_{m-1}-F_{m-1}^{2}$ $\displaystyle=$ $\displaystyle F_{m}^{2}-(F_{m}+F_{m-1})F_{m-1}$ $\displaystyle=$ $\displaystyle F_{m}^{2}-F_{m+1}F_{m-1}$ $\displaystyle=$ $\displaystyle-(-1)^{m}$

by the induction hypothesis. So we get $F_{m+2}F_{m}-F_{m+1}^{2}=(-1)^{m+1}$, and the result is thus true for $n=m+1$. The theorem now follows by induction.

Title proof of Cassini’s identity ProofOfCassinisIdentity 2013-03-22 14:44:40 2013-03-22 14:44:40 yark (2760) yark (2760) 24 yark (2760) Proof msc 11B39 CatalansIdentity