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  • If the arithmetic mean and geometric mean of the pth and qth terms of the sequence -16,8,-4,2..... satisfy the equation <img alt="4x^2-9x+5=0" src="https://entrancecorner.codecogs.com/gif.latex?4x%5E2-9x&plus;5%

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

Answers (1)

best_answer

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

 

Posted by

himanshu.meshram

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