** **

**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, _{}_{}.**