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Question Number 94397 by peter frank last updated on 18/May/20

Answered by mr W last updated on 18/May/20

$${(3x+5)}^{10}$$$$=5^{10}{(1+\frac{3x}{5})}^{10}$$$$=5^{10}\underset{k=0}{\sum ^{10}}{C}_k^{10}{\left(\frac{3x}{5}\right)}^k$$$$=5^{10}\underset{k=0}{\sum ^{10}}{C}_k^{10}{\left(\frac{3}{5}\right)}^k{x}^k$$$$let{a}_k=C_k^{10}{\left(\frac{3}{5}\right)}^k$$$$a_{k}ismaximumwhen$$$$a_{k-1}⩽a_k>a_{k+1}$$$$from{a}_k>a_{k+1}weget$$$$C_k^{10}{\left(\frac{3}{5}\right)}^k>C_{k+1}^{10}{\left(\frac{3}{5}\right)}^{k+1}$$$$\frac{10!}{k!(10-k)!}>\frac{10!}{(k+1)!(10-k-1)!}\times \frac{3}{5}$$$$\frac{1}{10-k}>\frac{1}{k+1}\times \frac{3}{5}$$$$30-3k<5k+5$$$$25<8k$$$$k>\frac{25}{8}\Rightarrow k⩾4$$$$i.e.a_{4}ismaximum.$$$$themax.coef.isof{x}^4:$$$$5^{10}\times C_4^{10}\times {\left(\frac{3}{5}\right)}^4=5^6\times 3^4\times 210$$$$=265781250$$

Commented by peter frank last updated on 19/May/20

$$thankyousir$$

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