A geometric progression consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying odd places, then the common ratio of the G.P is

Step-by-step solution

Let the geometric progression be a, ar, ar^2, ar^3, , ar^ 2n-1 , where the number of terms is 2n . The sum of all terms is S = a + ar + ar^2 + + ar^ 2n-1 = a r^ 2n - 1 r - 1 . The terms occupying odd places are a, ar^2, ar^4, , ar^ 2n-2 . This is a geometric progression with first term a , common ratio r^2 , and n terms. The sum of terms at odd places is S_ odd = a + ar^2 + ar^4 + + ar^ 2n-2 = a (r^2)^n - 1 r^2 - 1 = a r^ 2n - 1 r^2 - 1 . Given that S = 5 S_ odd , we have a r^ 2n - 1 r - 1 = 5 a r^ 2n - 1 r^2 - 1 .