# 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 $n\ge 2$.
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 $m\ge 2$.
Recalling the recurrence relation for the Fibonacci numbers,
${F}_{i+1}={F}_{i}+{F}_{i-1}$, we have

${F}_{m+2}{F}_{m}-{F}_{m+1}^{2}$ | $=$ | $({F}_{m+1}+{F}_{m}){F}_{m}-{({F}_{m}+{F}_{m-1})}^{2}$ | ||

$=$ | ${F}_{m+1}{F}_{m}+{F}_{m}^{2}-{F}_{m}^{2}-2{F}_{m}{F}_{m-1}-{F}_{m-1}^{2}$ | |||

$=$ | ${F}_{m+1}{F}_{m}-2{F}_{m}{F}_{m-1}-{F}_{m-1}^{2}$ | |||

$=$ | $({F}_{m}+{F}_{m-1}){F}_{m}-2{F}_{m}{F}_{m-1}-{F}_{m-1}^{2}$ | |||

$=$ | ${F}_{m}^{2}+{F}_{m-1}{F}_{m}-2{F}_{m}{F}_{m-1}-{F}_{m-1}^{2}$ | |||

$=$ | ${F}_{m}^{2}-{F}_{m}{F}_{m-1}-{F}_{m-1}^{2}$ | |||

$=$ | ${F}_{m}^{2}-({F}_{m}+{F}_{m-1}){F}_{m-1}$ | |||

$=$ | ${F}_{m}^{2}-{F}_{m+1}{F}_{m-1}$ | |||

$=$ | $-{(-1)}^{m}$ |

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 |
---|---|

Canonical name | ProofOfCassinisIdentity |

Date of creation | 2013-03-22 14:44:40 |

Last modified on | 2013-03-22 14:44:40 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 24 |

Author | yark (2760) |

Entry type | Proof |

Classification | msc 11B39 |

Related topic | CatalansIdentity |