Let an be the nth term of the G.P. of positive numbers.  Let ∑n=1100 a2n=α and ∑n=1100 a2n−1=β, such that α≠β, then the common ratio is

Let a be the first term and r, the common ratio of the  given G.P. Thenα−∑n=1100 a2n⇒α−a2+a4+…+a200⇒ α=ar+ar3+…+ar199⇒ α=ar1+r2+r4+…+r198        (1) and  β=∑n=1100 a2n−1⇒β=a1+a3+…+a199⇒ β=a+ar2+…+ar198              (2)⇒ β=a1+r2+…+r198From Eqs (1) and (2), we get   αβ=r.