denote the nth Fibonacci number. Prove that

denote the nth Fibonacci number. Prove that

∑ i=1ⁿ F²i = F(sub n)F(sub n+1) for every postive integer n.

2. Suppose that P(n) is a statement about n such that for any k ≥1 the truth of P(2k) implies the truth of P(2k + 2), and the truth of P(2k + 1) implies the truth of P(2k + 3). What (if any) base step/s would you prove in order to establish that P(n) is true for all positive integers?

3. Give a proof, following Steinhaus, that √17 is irrational.

4. For each pair of a and b, find the integers q and r with 0 ≤ r

(a) a = 37, b = 4
denote the nth Fibonacci number. Prove that