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Q.

If the arithmetic mean and geometric mean of the P^thand 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 ______

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Detailed Solution

$4 x^{2} - 9 x + 5 = 0 \Rightarrow x = 1, \frac{5}{4} Now, given \frac{5}{4} = \frac{t_{p} + t_{q}}{2}, t = t_{p} t_{q} where t_{r} = - 16 {(- \frac{1}{2})}^{r - 1} So \frac{5}{4} = - 8 [{(- \frac{1}{2})}^{p - 1} + {(- \frac{1}{2})}^{q - 1}] 1 = 256 {(- \frac{1}{2})}^{p + q - 2} \Rightarrow 2^{p + q - 2} = {(- 1)}^{p + q - 2} 2^{8} 1 = 256 {(- \frac{1}{2})}^{p + q - 2} \Rightarrow 2^{p + q - 2} = {(- 1)}^{p + q - 2} 2^{8}$

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