Sum of a geometric series:

The sum of the first 5 terms is twice the sum of the terms from the 6th to the 15th inclusive implies:

(note: the expression in the brackets is now a difference of two CUBES, so we can use:

So

Let

Using the quadratic formula ( ),

Putting back in:

But the question said that is a positive number, so must ALSO be positive, therefore

Proving that