4.7.2: Proofs by strong induction - explicit formulas for recurrence relations
Prove each of the following statements using strong induction:
(a) The Fibonacci sequence is defined as follows:
f0 = 0
f1 = 1
fn = fn-1 + fn-2, for n ≥ 2
Prove that for n ≥ 0, fn = 1/√5 [((1+√5)/2)^n - ((1-√5)/2)^n]
(b) Define the sequence {hn} as follows:
h0 = 5/3
h1 = 11/3
hn = 3hn-1 + 4hn-2 + 6n, for n ≥ 2
Prove that for n ≥ 0, hn = 2 ⋅ 4^n + 3/2(-1)^n - n - 11/6
(c) Define the sequence {gr} as follows: