# geometric series

By telescoping products
Assuming :

Sum of the first terms
1.

2.

So

and is the generating function for the sequence , where the first terms are .

3.

Multiply both sides by :

Subtract the second equation from the first:

Problem: Let , be two distinct primes. Prove that there are positive integers , so that the arithmetic mean of all the divisors of is also an integer.
Solution: The sum of all divisors of is given by

as can be seen by expanding the brackets. The number has positive divisors, and their arithmetic mean is

If and are both odd, then when and ,

If , choose and .
Then . (And similarly if ).