1. The problem statement, all variables and given/known data Show via induction that the nth root of (a1 * a2 * a3 * ... an) ≤ 1/ (n) * ∑ ai, where i ranges from 1 to n. 2. Relevant equations Induction 3. The attempt at a solution Let Pn be the statement above. It is clear that P1 holds since a1 ≤ a1. Now let us assume that Pn holds for any arbitrary integer k, that is the kth root of (a1 * a2 * a3 * a4 * ... ak) ≤ 1/k * ∑ ai where i ranges from 1 to k. I need to show that the (k + 1)th root is ≤ 1/ (k + 1) * ∑ ai, where i ranges from 1 to k + 1. I have had no such luck doing this. Would complete induction be required here? The source of the problem is from Abstract Algebra, Theory and Applications from T. W. Judson (2013 version).