JEE PYQ: Sequence And Series Question 17
Question 17 - 2021 (26 Feb Shift 2)
If the arithmetic mean and geometric mean of the $p^{th}$ and $q^{th}$ terms of the sequence $-16, 8, -4, 2, \ldots$ satisfy the equation $4x^2 - 9x + 5 = 0$, then $p + q$ is equal to
Show Answer
Answer: (10)
Solution
Given, $4x^2 - 9x + 5 = 0$
$\Rightarrow (x - 1)(4x - 5) = 0$
$\Rightarrow$ A.M $= \frac{5}{4}$, G.M $= 1$ (QA.M $>$ G.M)
Again, for the series $-16, 8, -4, 2, \ldots$
$p^{th}$ term $t_p = -16\left(-\frac{1}{2}\right)^{p-1}$
$q^{th}$ term $t_q = -16\left(-\frac{1}{2}\right)^{q-1}$
Now, A.M $= \frac{t_p + t_q}{2} = \frac{5}{4}$ & G.M $= \sqrt{t_p t_q} = 1$
$\Rightarrow 16^2\left(-\frac{1}{2}\right)^{p+q-2} = 1$
$\Rightarrow (-2)^8 = (-2)^{p+q-2}$
$\therefore p + q = 10$
BETA
