If the arithmetic mean and the geometric mean of the p th and q th terms of the sequence - 16 , 8 , - 4 , 2 , … satisfy the equation 4 x 2 - 9 x + 5 = 0 , then p + q is equal to _______.

Step-by-step solution

Let t p   &   t q is the p t h   &   q t h   terms of the series - 16 , 8 , - 4 , 2 , . . . Here a = - 16 ,   r = - 1 2 Then, t p = - 16 - 1 2 p - 1 t q = - 16 - 1 2 q - 1 The roots of the given quadratic equation 4 x 2 - 9 x + 5 = 0 is x = 1 , 5 4 .   A M = 5 4 ,   G M = 1   ∵ A M ≥ G M Now, given 5 4 = t p + t q 2 , 1 = t p t q So, 1 = 256 - 1 2 p + q - 2 ⇒ 2 p + q - 2 = - 1 p + q - 2 2 8 ⇒ 2 p + q - 10 = - 1 p + q - 2 It is equal when 2