Question

Recall that the sequence of Fibonacci numbers fn (n = 0, 1, 2, ...) is defined by the recursive equations f0 = 0 f1 = 1 fn = fn-1 + fn-2 for n ? 2. Prove each of the following properties of the Fibonacci numbers using an appropriate induction principle (induction, strong induction or well ordering). a. The inequality fn+1 < (13/8)^n holds for all n ? 1. b. Given a fixed number m ? 1 the number fmn is divisible by fm for all n ? 1. Hint prove this by induction on n. For the inductive step you may assume that the equality fr+m = fr-1fm + frfm+1 holds for all m ? 1 and r ? 1. c. The equation ?(i=0 to 2n+1) fifi+1 = f^2_{2n+2} holds for all natural numbers n ? 1.

          Recall that the sequence of Fibonacci numbers fn (n = 0, 1, 2, ...) is defined by the recursive equations

f0 = 0
f1 = 1
fn = fn-1 + fn-2 for n ? 2.

Prove each of the following properties of the Fibonacci numbers using an appropriate induction principle (induction, strong induction or well ordering).

a. The inequality fn+1 < (13/8)^n holds for all n ? 1.
b. Given a fixed number m ? 1 the number fmn is divisible by fm for all n ? 1.
Hint prove this by induction on n. For the inductive step you may assume that the equality fr+m = fr-1fm + frfm+1 holds for all m ? 1 and r ? 1.
c. The equation ?(i=0 to 2n+1) fifi+1 = f^2_{2n+2} holds for all natural numbers n ? 1.

Recall that the sequence of Fibonacci numbers fn (n = 0, 1, 2, ...) is defined by the recursive equations

f0 = 0
f1 = 1
fn = fn-1 + fn-2 for n ? 2.

Prove each of the following properties of the Fibonacci numbers using an appropriate induction principle (induction, strong induction or well ordering).

a. The inequality fn+1 < (13/8)^n holds for all n ? 1.
b. Given a fixed number m ? 1 the number fmn is divisible by fm for all n ? 1.
Hint prove this by induction on n. For the inductive step you may assume that the equality fr+m = fr-1fm + frfm+1 holds for all m ? 1 and r ? 1.
c. The equation ?(i=0 to 2n+1) fifi+1 = f^22n+2 holds for all natural numbers n ? 1.

Added by Michele G.

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