Fibonacci Sequence Induction Problem

[blak97]

Homework Statement

Show that for all n greater than 1:

f_n = $\frac{1}{\sqrt{5}}${($\frac{1+\sqrt{5}}{2}$)ⁿ - ($\frac{1-\sqrt{5}}{2}$)ⁿ}

Homework Equations

f₁ = f₂= 1
f_n+2 = f_n+1 + f_n

The Attempt at a Solution

I'm pretty sure it's by induction, but I'm not sure how to start.

 

[vela]

What do you need in a proof by induction?

 

[blak97]

What do you mean by what do I need?

 

[vela]

What's the basic structure of an induction proof? You should be able to at least start the proof.

 

[blak97]

Well what I have done so far is input n+1 and n into the given expression to give:
f_n+1 + f_n = $\frac{1}{\sqrt{5}}$ {[$\frac{1+\sqrt{5}}{2}$]ⁿ⁺¹ - [$\frac{1-\sqrt{5}}{2}$]ⁿ⁺¹ + [$\frac{1+\sqrt{5}}{2}$]ⁿ - [$\frac{1-\sqrt{5}}{2}$]ⁿ}

I need to make this equal to (in order to prove by induction):
f_n+2 = $\frac{1}{\sqrt{5}}$ {[$\frac{1+\sqrt{5}}{2}$]ⁿ⁺² - [$\frac{1-\sqrt{5}}{2}$]ⁿ⁺²}

 

[vela]

Hint: What's $\big(\frac{1\pm\sqrt{5}}{2}\big)^2$?

 

[blak97]

Does it have something to do with the quadratic formula?