If the arithmetic mean and geometric mean of the $\mathrm{p}^{\mathrm{th}}$ and $\mathrm{q}^{\text {th }}$ terms of the sequence $-16,8,-4,2, \ldots$ satisfy the equation $4 x^{2}-9 x+5=0$, then $p+q$ is equal to
$4 x^{2}-9 x+5=0 \Rightarrow x=1, \frac{5}{4}$
Now given $\frac{5}{4}=\frac{\mathrm{t}_{\mathrm{p}}+\mathrm{t}_{\mathrm{q}}}{2}, \mathrm{t}=\mathrm{t}_{\mathrm{p}} \mathrm{t}_{\mathrm{q}}$ where
$t_{r}=-16\left(-\frac{1}{2}\right)^{r-1}$
so $\frac{5}{4}=-8\left[\left(-\frac{1}{2}\right)^{p-1}+\left(-\frac{1}{2}\right)^{q-1}\right]$
$1=256\left(-\frac{1}{2}\right)^{\mathrm{p}+\mathrm{q}-2} \Rightarrow 2^{\mathrm{p}+\mathrm{q}-2}=(-1)^{\mathrm{p}+\mathrm{q}-2} 2^{8}$
hence $p+q=10$
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