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  • <img alt="\text { For } x \in R, x \neq-1 \text {, if }(1+x)^{2016}+x(1+x)^{2015}+x^2(1+x)^{2014}+" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Ctext%20%7B%20For%20%7D%20x%20%5Cin%20R%2C

\text { For } x \in R, x \neq-1 \text {, if }(1+x)^{2016}+x(1+x)^{2015}+x^2(1+x)^{2014}+ \ldots .+x^{2016}=\sum_{i=0}^{2016} a_i x^i, \text { then } a_{2016} \text { is equal to }

Option: 1

2019


Option: 2

2018


Option: 3

2016


Option: 4

2017


Answers (1)

best_answer

\text { Let } S=(1+x)^{2016}+x(1+x)^{2015}+x^2(1+x)^{2014}+\ldots .+x^{2015}(1+x)+x^{2016}...(i)

\begin{aligned} \left(\frac{x}{1+x}\right) S=x(1+x)^{2015}+x^2(1+x)^{2014}+\ldots & \\ & +x^{2016}+\frac{x^{2017}}{1+x} \end{aligned}...(ii)

Subtracting (ii) from (i), we get

\begin{aligned} & \frac{S}{1+x}=(1+x)^{2016}-\frac{x^{2017}}{1+x} \Rightarrow S=(1+x)^{2017}-x^{2017} \\ & \therefore \quad a_{2016}=\text { coefficient of } x^{2016}={ }^{2017} C_{2016}=2017 \end{aligned}

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manish painkra

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