Abstract Algebra Proof by induction problem

[Isaac Wiebe]

Homework Statement

Show via induction that the nth root of (a1 * a2 * a3 * ... an) ≤ 1/ (n) * ∑ ai, where i ranges from 1 to n.

Homework Equations

Induction

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).

 

[Ray Vickson]

Homework Statement

Show via induction that the nth root of (a1 * a2 * a3 * ... an) ≤ 1/ (n) * ∑ ai, where i ranges from 1 to n.

Homework Equations

Induction

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).

Are all the a_i supposed to be > 0? If so, try first to look at the simple case of n = 2.

 

[Isaac Wiebe]

All ai ⋲ N, so yes they are. And for n = 2 I eventually receive that √(a1 * a2) ≤ 1/2 (a1 + a2)
Or a1 * a2 ≤ [(a1 + a2)^2] / 4. Not entirely sure why I would want to do multiple base cases, but I think you are on the right track.