So my base cases check to be true. I've attached my work here.
And fn is true for n. But I'm not sure how to prove for n + 1.
fn+1= f(n+1)-1 + f(n+1)-2
=fn+f(n-1)
=1/sqrt(5)[ϕn+ϕ-n] + 1/sqrt(5) [ϕ(n-1) ϕ-(n-1)
What do I do next? If I'm even going in the right direction.
Recall that the fibonacci sequence is defined as
{ f0=0; f1 = 1 and
{fn = f n - 1 + fn -2 for n 2
Prove by generalized mathematical induction that
fn = 1/sqrt(5)[ϕn - (-ϕ)-n]
where ϕ = [1+sqrt(5)]/2
is the golden ratio.. (This is known as de Moivre's formula.)
So...