If n is not a prime number, then n=ab ,
For some positive integers a, b>1, we get
So, n is composite and 2n−1 is also composite which is a contradiction.
Thus, n is prime
Let an be the nth term of the G.P. of positive numbers. Let ∑100n=1a2n=α and ∑100n=1a2n−1=β, such that such that α≠β. Prove that the common ratio of the G.P. is αβ.