Question

The coefficient of $x^n$ in the expansion of $\frac{e^{7x}+e^x}{e^{3x}}$ is

• A. $\frac{4^{n-1}-{(-2)}^{n-1}}{n!}$
• B. $\frac{4^{n-1}-{\left(2\right)}^{n-1}}{n!}$
• C. $\frac{4^n+{(-2)}^n}{n!}$
• D. $\frac{4^{n-1}-{(-2)}^n}{n!}$

Answer 1

The correct option is C $\frac{4^n+{(-2)}^n}{n!}$

$\frac{e^{7x}+e^x}{e^{3x}}=e^{4x}+e^{-2x}$
$\therefore$ Coefficient of $x^n$ in $(e^{4x}+e^{-2x})$
$=\frac{{\left(4\right)}^n}{n!}+\frac{{(-2)}^n}{n!}$