Problem 10

 

Suppose  are positive real numbers satisfying  for all   If k is any integer, find the sum in terms of n only (the sum turns out to be independent of k) 

. 

[Problem submitted by Iris Magee, LACC Instructor of Mathematics.]

 

Solution:

 

Let  stand for the sum:  = .

Now rewrite the sum in reverse order:   =

We were given that .  This implies

So, .

Add the first two equations to get

2=   Therefore,  .