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Homeproof of Cassini's identity

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# 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.

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