(Solution) - A Let f Z R where f n ni 1 -(2025 Original AI-Free Solution)

Paper Details

a) Let f: Z+ ? R where f(n) = ?ni=1 1. When n = 4, for example, we have f(n) = f(4) = 1 + 2 + 3 + 4 > 2 + 3 + 4>2 + 2 + 2 = 3- 2 = [(4+ l)/2]2 = 6 > (4/2)2 = (n/2)1. For n = 5, we find f(n) = /(5) = l+2 + 3 + 4 + 5>3 + 4 + 5 > 3 + 3 + 3 = 3? 3 = [(5 + 1)(n/2)|3 = 9 > (5/2)2 = (n/2)2. In general, f(n) = 1 + 2 +--------+ n > [n/2] + ??????? + n = [n/2] + ?????? + ] = [n/2 = [(n/2)] = [(n + 1)/2][n/2] > n2/4
Consequently, f e ?(n2).
Use

to provide an alternative proof that f ? ?(n2).
(b) Let g: Z+ ? R where g(n) = ?ni=1 i2- Prove that g ? ?(n3).
(c) For t ? Z+, let h: Z+ ? R where h(n) = ?ni=1 it.
Prove that h ? ?(nt+1).