In an infinite geometric progression, the sum of all the even-positioned terms is 1 4 of the sum of the entire progression. If the first term of the progression is 4 , then the sum of the squares of all the terms of the progression is equal to

Step-by-step solution

Let the infinite geometric progression be a, ar, ar^2, ar^3, The sum of the entire progression is S_ total = a 1 - r . The sum of the even-positioned terms is S_ even = ar + ar^3 + ar^5 + = ar 1 - r^2 . Given that S_ even = 1 4 S_ total , we can write: S_ total = S_ odd + S_ even S_ odd = S_ total - 1 4 S_ total = 3 4 S_ total . Since each even-positioned term is r times the preceding odd-positioned term, we have S_ even = r S_ odd . r = S_ even S_ odd = 1 4 S_ total 3 4 S_ total = 1 3 . We are given the first term