Let the first term a and the common ratio r of a geometric progression be positive integers with r > 1 . If the sum of its first four terms is 280 , then the sum of the third and fifth terms of this progression is equal to

Step-by-step solution

Let the geometric progression be a, ar, ar^2, ar^3, The sum of the first four terms is given by a + ar + ar^2 + ar^3 = 280 a(1 + r + r^2 + r^3) = 280 a(1 + r)(1 + r^2) = 280 Since a and r are positive integers and r > 1 , we can check integer values for r : For r = 2 , (1 + r)(1 + r^2) = (3)(5) = 15 . Since 280 is not divisible by 15 , r 2 . For r = 3 , (1 + r)(1 + r^2) = (4)(10) = 40 . Here, 280 is divisible by 40 , giving a = 280 40 = 7 . For r 4 , the factor (1 + r)(1 + r^2) (5)(17) = 85 . The only multiple of 8