The sum of four numbers in a geometric progression is 60 , and the arithmetic mean of the first and the last number is 18 . Then the numbers are

Step-by-step solution

Let the four numbers in geometric progression be a, ar, ar^2, ar^3 . The sum of the four numbers is a + ar + ar^2 + ar^3 = 60 . This can be written as a(1 + r + r^2 + r^3) = 60 , or a(1 + r)(1 + r^2) = 60 . The arithmetic mean of the first and last number is a + ar^3 2 = 18 , which implies a(1 + r^3) = 36 . Since 1 + r^3 = (1 + r)(1 - r + r^2) , we have a(1 + r)(1 - r + r^2) = 36 . Dividing the two equations: a(1 + r)(1 + r^2) a(1 + r)(1 - r + r^2) = 60 36 = 5 3 . This simplifies to 1 + r^2 1 - r + r^2 = 5 3 . Cros