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Q.
The co-efficient of $x^{n}$ in the expansion of $\frac{ e ^{7 x }+ e ^{ x }}{ e ^{3 x }}$ is
Binomial Theorem
Solution:
$\frac{ e ^{7 x }+ e ^{ x }}{ e ^{3 x }}= e ^{4 x }+ e ^{-2 x }$
$=\left[1+4 x +\frac{(4 x )^{2}}{2 !}+\ldots . .\right]+\left[1+(-2 x )+\frac{(-2 x )^{2}}{2 !}+\ldots \ldots\right]$
$\therefore $ coeff. of $x^{n}=\frac{4^{n}}{n !}+\frac{(-2)^{n}}{n !}$