Answer Happy

Let fn be the nth Fibonacci number. Prove the following by induction on n € Z+. (i) Σ₁_₁f² = fnfn+1 (ii) 2₁ fi-fi = f'2m (iii) f₂ + f2+1 = f2n+1 [Hint: For any n ≥ 2, we can write the nth Fibonacci number as fn = fkfn-k-1 + fk+1fn-k for any k = 1,2,...,n-2.]