The product of three numbers in geometric progression is 8 and the sum of the product of the numbers taken in pairs is 14. Find the numbers.

Step-by-step solution

Let the three numbers in geometric progression be a r , a, ar . Given that the product of the numbers is 8 : a r a ar = 8 a^3 = 8 a = 2 Given that the sum of the product of the numbers taken in pairs is 14 : a r a + a ar + a r ar = 14 a^2 r + a^2 r + a^2 = 14 Substituting a = 2 : 4 r + 4r + 4 = 14 4 ( 1 r + r ) = 10 1 r + r = 5 2 2r^2 - 5r + 2 = 0 (2r - 1)(r - 2) = 0 r = 2 or r = 1 2 For r = 2 , the numbers are 1, 2, 4 . For r = 1 2 , the numbers are 4, 2, 1 . Answer: 4, 2, 1 and 1, 2, 4