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Question Number 100587 by bobhans last updated on 27/Jun/20

$$\mathrm{If}\mathrm{the}\mathrm{coefficients}\mathrm{of}{x}^k\mathrm{and}{x}^{k+1}\mathrm{in}\mathrm{the}$$$$\mathrm{expansion}{(2+3x)}^{19}\mathrm{are}\mathrm{equal},\mathrm{what}\mathrm{is}$$$$\mathrm{the}\mathrm{value}\mathrm{of}k?$$

Commented by bobhans last updated on 27/Jun/20

$$\mathrm{thank}\mathrm{you}\mathrm{both}$$

Answered by maths mind last updated on 27/Jun/20

$${(2+3x)}^{19}=\mathrm{\Sigma }{C}_{19}^k{\left(3x\right)}^k{2}^{19-k}$$$$\Rightarrow C_{19}^m{3}^m{2}^{19-m}=C_{19}^{m+1}{3}^{m+1}{2}^{19-m-1}\Rightarrow$$$$\frac{C_{19}^m}{C_{19}^{m+1}}=\frac{3}{2}$$$$\Rightarrow \frac{(m+1)!.(18-m)!}{m!.(19-m)!}=\frac{3}{2}\iff 2(m+1)=3(19-m)$$$$\Rightarrow 5m=55\Rightarrow m=11$$

Answered by john santu last updated on 27/Jun/20

$$\mathrm{by}\mathrm{Binomial}\mathrm{Theorem}$$$${(2+3x)}^{19}=\underset{\mathrm{n}=0}{\sum ^{19}}\left(\begin{array}{c}19\\ \mathrm{n}\end{array}\right){\left(2\right)}^{19-\mathrm{n}}{\left(3x\right)}^n$$$$wewanttofindksothat$$$$\left(\begin{array}{c}19\\ k\end{array}\right){\left(2\right)}^{19-k}{\left(3x\right)}^k=\left(\begin{array}{c}19\\ k+1\end{array}\right){\left(2\right)}^{18-k}{\left(3x\right)}^{k+1}$$$$\Rightarrow 2.\frac{19!}{k!(19-\mathrm{k})!}=3.\frac{19!}{(\mathrm{k}+1)!(18-\mathrm{k})!}$$$$2.\frac{19!}{(19-\mathrm{k}).\mathrm{k}!(18-\mathrm{k})!}=3.\frac{19!}{(\mathrm{k}+1).\mathrm{k}!(18-\mathrm{k})!}$$$$\frac{2}{19-\mathrm{k}}=\frac{3}{\mathrm{k}+1}\iff k=11◼$$

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