# If the arithmetic mean and geometric mean of the pth and qth terms of the sequence -16,8,-4,2..... satisfy the equation $4x^2-9x+5=0$, then p+q is equal to ________ Option: 1 8 Option: 2 10 Option: 3 12 Option: 5 14

given sequence is,

-16, 8, -4, 2 ...

a = -16, r = -½

$\\T_{p}=-16\left(-\frac{1}{2}\right)^{p-1}=-16 \times-2 \times\left(-\frac{1}{2}\right)^{p}=32 \times\left(-\frac{1}{2}\right)^{p} \\ T_{q}=-16\left(-\frac{1}{2}\right)^{q-1}=32 \times\left(-\frac{1}{2}\right)^{q}$

\begin{aligned} &\frac{T_{p}+T_{q}}{2}=\text { A.M. and } \sqrt{T_{p}+T_{q}}=G . M\\ &\text { A.M. and G.M. are the roots of } 4 x^{2}-9 x+5=0 \end{aligned}

$\\\text { A.M. }=\frac{32 \times\left(\frac{-1}{2}\right)^{\mathrm{p}}+32 \times\left(\frac{-1}{2}\right)^{\mathrm{q}}}{2}=16\left[\left(\frac{-1}{2}\right)^{\mathrm{p}}+\left(-\frac{1}{2}\right)^{\mathrm{q}}\right] \\ \text { G.M. }=\sqrt{32 \times\left(\frac{-1}{2}\right)^{\mathrm{P}}\times32 \times\left(\frac{-1}{2}\right)^{\mathrm{q}}}=32 \sqrt{\left(\frac{-1}{{2}}\right)^{\mathrm{p}+\mathrm{q}}}$

$4 x^{2}-9 x+5=0 \Rightarrow x=1, \frac{5}{4}$

$\\32 \sqrt{\left(\frac{-1}{2}\right)^{p+q}}=1\\p+q=10$

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