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Question Number 116917 by I want to learn more last updated on 07/Oct/20

$$\mathrm{Find}\mathrm{the}\mathrm{greatest}\mathrm{coefficient}\mathrm{and}\mathrm{greatest}\mathrm{term}\mathrm{in}$$$${(3\mathrm{x}-2)}^{-7}.$$$$$$$$\mathrm{Sir}\mathrm{is}\mathrm{it}:{(-1)}^{-7}.{(2-3\mathrm{x})}^{-7}=-{(2-3\mathrm{x})}^{-7}$$$$=-\frac{8008\times 2^{10}}{3^{17}}$$

Answered by mr W last updated on 08/Oct/20

$${(3x-2)}^{-7}$$$$=-2^{-7}{(1-\frac{3x}{2})}^{-7}$$$$=-2^{-7}\underset{k=0}{\sum ^{\mathrm{\infty }}}{C}_6^{k+6}{\left(\frac{3}{2}\right)}^k{x}^k$$$$a_k=-2^{-7}{C}_6^{k+6}{\left(\frac{3}{2}\right)}^k$$$$alltermshavenegativecoefficients,$$$$whicharedecreasing,i.e.$$$$a_0>a_1>a_2>....$$$$thegreatestis{a}_0=-2^{-7}=-\frac{1}{128}$$

Commented by I want to learn more last updated on 08/Oct/20

$$\mathrm{Ohh},\mathrm{i}\mathrm{get}\mathrm{sir},\mathrm{this}\mathrm{makes}\mathrm{it}\mathrm{different}\mathrm{from}{(2-3\mathrm{x})}^{-7}\mathrm{which}\mathrm{is}\frac{8008\times 2^{10}}{3^{17}}.$$$$\mathrm{Thanks}\mathrm{sir},\mathrm{understand}.$$

Commented by mr W last updated on 08/Oct/20

$$C_6^{k+6}{\left(\frac{2}{3}\right)}^{k}hasamaximum.$$$$C_6^{k+6}{\left(\frac{3}{2}\right)}^{k}hasnomaximum.$$

Commented by I want to learn more last updated on 08/Oct/20

$$\mathrm{I}\mathrm{appreciate}\mathrm{sir}.$$

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