In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is R -(a, b) , then a²+b² is equal to _ _ _ _ .

Step-by-step solution

In the G.P., let the first three terms be a r , a, ar . From the product condition: a r a ar = a^3 = 27 , so a = 3 . The sum of first three terms is S = 3 r + 3 + 3r = 3( 1 r + 1 + r) . For r > 0 , by AM-GM: 1 r + r 2 , so S 9 . For r So S -3 . Thus the possible values of S are (- , -3] [9, ) = R - (-3, 9) . Therefore (a, b) = (-3, 9) and a^2 + b^2 = 9 + 81 = 90 .