## You are here

Homeproof of Cassini's identity

## Primary tabs

# 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.

Related:

CatalansIdentity

Type of Math Object:

Proof

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

11B39*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Jul 5

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias

Jun 13

new question: young tableau and young projectors by zmth

Jun 11

new question: binomial coefficients: is this a known relation? by pfb

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias

Jun 13

new question: young tableau and young projectors by zmth

Jun 11

new question: binomial coefficients: is this a known relation? by pfb