(Solved): (a) Suppose that c1, c2, c3, ... is a sequence such that c0 = 1, c1 = 1 and ck = ck/2 + ck/ ...
(a) Suppose that c1, c2, c3, ... is a sequence such that c0 = 1, c1 = 1 and ck = c⌊k/2⌋ + c⌈k/2⌉ for every integer k ≥ 2. Prove that cn = n for every integer n ≥ 1. (b) Prove that √ n < 1 √ 1 + 1 √ 2 + · · · + 1 √ n