If the coefficients of $x^5$ and $x^6$ are equal in the expansion of $\left(a+\frac{x}{5}\right)^{65}$, then the coefficient of $x^2$ in the expansion of $\left(a+\frac{x}{5}\right)^4$ is

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Aicharya · Accepted Answer

Coefficient of $x^5={ }^{65} \mathrm{C}_5 a^{60}\left(\frac{1}{5}\right)^5={ }^{65} \mathrm{C}_5 \frac{a^{60}}{5^5}$ Coefficient of $x^6={ }^{65} \mathrm{C}_6\left(\frac{1}{5}\right)^6 a^{59}={ }^{65} \mathrm{C}_6 \frac{a^{59}}{5^6}$ $\begin{aligned} & { }^{{ }^6 C_5} \frac{a^{60}}{5^5}={ }^{65} \mathrm{C}_6 \frac{a^{59}}{5^6} \\ & a=\frac{{ }^{65} \mathrm{C}_6}{{ }^{65} \mathrm{C}_5 \times 5}=\frac{5!60!}{59!6!} \times \frac{1}{5} \Rightarrow a=2 \\ & \left(a+\frac{x}{5}\right)^4=\left(2+\frac{x}{5}\right)^2 \end{aligned}$
Coefficient of $x^2$ is ${ }^4 C_2 \times 2^2 \times\left(\frac{1}{5}\right)^2=\frac{4!}{2!2!} \times \frac{2^2}{5^2}=\frac{24}{25}$

If the coefficients of $x^5$ and $x^6$ are equal in the expansion of $\left(a+\frac{x}{5}\right)^{65}$, then…

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