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  4. The coefficient of xk in the expansion of 1+(1 + x)+(1 + x)2+ ldots +(1 + x)n is equal to

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Q. The coefficient of $x^{k}$ in the expansion of $1+\left(1 + x\right)+\left(1 + x\right)^{2}+\ldots +\left(1 + x\right)^{n}$ is equal to

NTA AbhyasNTA Abhyas 2022

Solution:

$1+\left(1 + x\right)+\left(1 + x\right)^{2}+\ldots +\left(1 + x\right)^{n}=\frac{1}{x}\left[\left(1 + x\right)^{n + 1} - 1\right]$
Hence, required coefficient is ${}_{}^{n + 1}C_{k + 1}^{}$