If the arithmetic mean and geometric mean of the pth and qth terms of the
sequence $-$16, 8, $-$4, 2, ...... satisfy the equation
4x2 $-$ 9x + 5 = 0, then p + q is equal to __________.
Answer (integer)
10
Solution
Given, $4{x^2} - 9x + 5 = 0$<br><br>$\Rightarrow (x - 1)(4x - 5) = 0$<br><br>$\Rightarrow$ A. M. $= {5 \over 4}$, G. M. = 1 (As A. M. $\ge$ G. M)<br><br>Again, for the series<br><br>$-$16, 8, $-$4, 2 ..........<br><br>${p^{th}}$ term ${t_p} = - 16{\left( {{{ - 1} \over 2}} \right)^{p - 1}}$<br><br>${q^{th}}$ term ${t_p} = 16{\left( {{{ - 1} \over 2}} \right)^{q - 1}}$<br><br>Now, A. M. = ${{{t_p} + {t_q}} \over 2} = {5 \over 4}$ & G. M. = $\sqrt {{t_p}{t_q}} = 1$<br><br>$\Rightarrow {16^2}{\left( { - {1 \over 2}} \right)^{p + q - 2}} = 1$<br><br>$\Rightarrow {( - 2)^8} = {( - 2)^{(p + q - 2)}}$<br><br>$\Rightarrow p + q = 10$
About this question
Subject: Mathematics · Chapter: Sequences and Series · Topic: Arithmetic Progression
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