Induction

rmotome

I am looking for a write up of the proof of Binet's Formula using induction. I am learning to do proofs.
$F_{n}=(\alpha^{n}-\beta^{n})\sqrt{5}$
$\alpha=(1+\sqrt{5})/2$
$\beta=(1-\sqrt{5})/2$
Thanks, in advance
Robert Otome

**Posted:** Wed Aug 19, 2009 4:13 pm

DaVe89 · New Member Offline Joined: 10 Apr 2008 Posts: 7

First, Binet's formula is $F_{n}=\frac{(\alpha^{n}-\beta^{n})}{\sqrt{5}}=\frac{(1+\sqrt{5})^n-(1-\sqrt{5})^n}{2^n \sqrt{5}}$

It's true for n=1 and n=2.

Suppose that the equality is true for $n=k$ and $n=k+1$ , $k>2,k \in \mathbb{N}$ .

Then:
$F_{k+2}=F_{k}+F_{k+1}=\frac{(1+\sqrt{5})^k-(1-\sqrt{5})^k}{2^k \sqrt{5}}+\frac{(1+\sqrt{5})^{k+1}-(1-\sqrt{5})^{k+1}}{2^{k+1}...$

It has now been proved by mathematical induction that Binet's Formula holds for $\forall n \in \mathbb{N}$

**Posted:** Mon Aug 24, 2009 3:09 pm