Question

If the arithmetic mean and geometric mean of the p^th and q^th terms of the sequence -16, 8, -4, 2, ... satisfy the equation 4x² - 9x + 5 = 0, then p + q is equal to ..........

Answer 1

Answer is 10

4x² - 9x + 5 = 0

⇒ x = 1, 5/4

Now given 5/4 =\(\frac{t_p+t_q}2,\) t_p t_q where

t_r = -16 (-1/2)^r-1

so 5/4 = -8[(-1/2)^p-1+ (-1/2)^q-1]

1 = 256(-1/2)^p+q-2

^⇒ 2^p+q-2 = (-1)^p+q-2 2⁸

hence p + q =10