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

Step-by-step solution

Let the geometric progression have 2n terms, given by a, ar, ar^2, ar^3, , ar^ 2n-1 . The sum of all terms S is given by S = a(r^ 2n - 1) r - 1 . The terms occupying the 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 in odd places S_ odd is given by S_ odd = a((r^2)^n - 1) r^2 - 1 = a(r^ 2n - 1) r^2 - 1 . According to the problem, S = 5 S_ odd . Substituting the expressions: a(r^ 2n - 1) r - 1 = 5 a(r^ 2n - 1) r^2 - 1 .