# Fibonacci sequence in closed form

Prove that Fn can be expressed by:

Fn=[(a^n-b^n)/(a-b)] for n=1,2,3...

~: Set Gn=[(a^n-b^n)/(a-b)] and show that Gn satisfies the recurrence formula {G(n+1) = Gn + G(n-1) for n=2,3,4...} and don't forget that a and b satisfy the equation x^2-x-1=0.

a and b are the roots of x^2-x-1=0, which are (1+sqrt5)/2 and (1-sqrt5)/2.

https://brainmass.com/math/recurrence-relation/fibonacci-sequence-closed-form-10310

#### Solution Preview

Please see attached file.

This can be easily proved by Mathematical Induction:

We show that the recurrence relationship holds for n = 2 and n=3 by direct substitution:

At n =1 , G(1) =

At n = 2, G(2) = = a + b = 1 (plugging in a ...

#### Solution Summary

The Fibonacci sequence in closed forms are given. The recurrence relation is examined.

$2.19