The coefficient of x^101 in the expression (5+x)^500+x(5+x)^499+x^2(5+x)^498++x^500, x0, …

Question

The coefficient of $x^{101}$ in the expression $(5+x)^{500}+x(5+x)^{499}+x^2(5+x)^{498}+\cdots+x^{500}$, $x>0$, is

Accepted Answer

This is a G.P. with first term $(5+x)^{500}$, common ratio $\dfrac{x}{5+x}$, and 501 terms:

$$\text{Sum}=(5+x)^{500}\cdot\frac{\left(\frac{x}{5+x}\right)^{501}-1}{\frac{x}{5+x}-1}=\frac15\left[(5+x)^{501}-x^{501}\right]$$

Coefficient of $x^{101}$ in $(5+x)^{501}$ is ${}^{501}C_{101}\,5^{400}$.

$$\text{Coefficient of }x^{101}\text{ overall}=\frac15\times{}^{501}C_{101}\,5^{400}={}^{501}C_{101}\,5^{399}$$

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