$\frac{x}{(1-4 x)(1-5 x)}$ - এর বিস্তৃতিতে $x^{r}$ এর সহগ $\left(5^{r}+k 4^{r}\right)$ হলে k এর মান কত?

$\frac{x}{(1-4 x)(1-5 x)}$ $=\frac{\frac{1}{4}}{(1-4 x)\left(1-\frac{5}{4}\right)}$ $+\frac{\frac{1}{5}}{\left(1-\frac{4}{5}\right)(1-5 x)}$ $=-\frac{1}{1-4 x}+\frac{1}{1-5 x}$ $=(1-5 x)^{-1}-(1-4 x)^{-1}$ $\Rightarrow\left(1+5 x+\cdots+5^{r} x^{r}+\cdots\right)$ $-\left(1+4 x+\cdots 4^{r} x^{r}+\cdots\right)$ $\therefore \mathrm{x}^{\mathrm{r}}$ এর সহগ $5^{r}-4^{r}$ প্রশ্নমতে $5^{r}+k 4^{r}=5^{r}-4^{r}$ $\Rightarrow k=-1$