For x R , the expressions 3^ ^2 x + 3^ ^2 x , K 2 , 9^ ^2 x + 9^ ^2 x form three consecutive terms of an A.P. Let S be the set of all possible values of K . The sum of all integral values in the set S is :

Step-by-step solution

Since the three terms are in A.P., we have: 2 ( K 2 ) = 3^ ^2 x + 3^ ^2 x + 9^ ^2 x + 9^ ^2 x K = 3^ ^2 x + 3^ 1 - ^2 x + (3^ ^2 x )^2 + (3^ 1 - ^2 x )^2 Let t = 3^ ^2 x . Since 0 ^2 x 1 , we have 3^0 t 3^1 , so t [1, 3] . Substituting t into the equation for K gives: K = t + 3 t + t^2 + 9 t^2 Let u = t + 3 t . We find the range of u for t [1, 3] . The minimum of u occurs when t = 3 , giving u = 3 + 3 3 = 2 3 . The maximum of u occurs at the boundaries t = 1 or t = 3 , giving u = 1 + 3 = 4 . Thus, u [2 3 , 4] . Now