Given:

alpha = (1+ sqrt5)/2 and beta = (1-sqrt5)/2

alpha^2 = 1 + alpha and beta^2 = 1+ beta

Use induction to prove that for all integers n >= 1 we have

alpha^n = f(n-1)+ f(n)*(alpha) and beta^n = f(n-1)+ f(n)*(beta)

I've did my base case and plug in k+1 to n but I can't get them equal to each other. Please help, I've been playing with these numbers for hours.