Verified Solution • 2025–26 Session moderate

QJEEM21Ma88d6: JEE Main 2021 Mathematics - Class 11 JEEM21Ma88d6 Solution: If the arithmetic mean and geometric mean of the p th and q th…

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 __________.

Step-by-Step Verified Solution

Given, 4x2−9x+5=04{x^2} - 9x + 5 = 0

⇒(x−1)(4x−5)=0\Rightarrow (x - 1)(4x - 5) = 0

⇒\Rightarrow A. M. =54= {5 \over 4} , G. M. = 1 (As A. M. ≥\ge G. M)

Again, for the series

−- 16, 8, −- 4, 2 ..........

pth{p^{th}} term tp=−16(−12)p−1{t_p} = - 16{\left( {{{ - 1} \over 2}} \right)^{p - 1}}

qth{q^{th}} term tp=16(−12)q−1{t_p} = 16{\left( {{{ - 1} \over 2}} \right)^{q - 1}}

Now, A. M. = tp+tq2=54{{{t_p} + {t_q}} \over 2} = {5 \over 4} & G. M. = tptq=1\sqrt {{t_p}{t_q}} = 1

⇒162(−12)p+q−2=1\Rightarrow {16^2}{\left( { - {1 \over 2}} \right)^{p + q - 2}} = 1

⇒(−2)8=(−2)(p+q−2)\Rightarrow {( - 2)^8} = {( - 2)^{(p + q - 2)}}

⇒p+q=10\Rightarrow p + q = 10

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