Question

In the expansion of $\frac{(e^{7x}+e^{3x})}{e^{5x}}$, the constant term is

• A. $0$
• B. $1$
• C. $2$
• D. None of these

Answer 1

The correct option is C

$2$

Determine the constant term in $\frac{(e^{7x}+e^{3x})}{e^{5x}}$

$$\begin{gathered} \Rightarrow \frac{e^{7x}}{e^{5x}}+\frac{e^{3x}}{e^{5x}} \\ \Rightarrow e^{2x}+e^{-2x} \\ \Rightarrow \left(1+2x+\frac{4x^2}{2}+\frac{8x^3}{3}+\frac{16x^4}{4}+....\right)+\left(1-2x+\frac{4x^2}{2}-\frac{8x^3}{3}+\frac{16x^4}{4}-....\right)\left[\because e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+......\right] \\ \Rightarrow \left(1+1\right)[\text{Neglecting other terms}] \\ \Rightarrow 2 \end{gathered}$$Hence, option C is correct .