Question

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

• (A) $$\frac{4^{n-1}-(-2{)}^{n-1}}{\lfloor n}$$
• (B) $$\frac{4^{n-1}-2^{n-1}}{\lfloor n}$$
• (C) $$\frac{4^n-2^n}{\lfloor n}$$
• (D) $$\frac{4^n+(-2{)}^n}{\lfloor n}$$

Answer 1

The Correct Option is D

Explanation:
$$\frac{e^{7x}+e^{\pi }}{e^{3x}}=e^{4x}+e^{-2x}$$Co-efficient of $x^n$$$\frac{(4{)}^n}{n!}+\frac{(2{)}^n}{n!}(-1{)}^n=\frac{4^n+(-2{)}^n}{n!}$$