Question

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

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

Answer 1

Answer: (C)

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