The co-efficient of xn in the expansion of ( e 7 x + e x / e 3 x ) is

Question

The co-efficient of xn in the expansion of ( e 7 x + e x / e 3 x ) is (A) (4 n -1+(-2) n / n !) (B) (4 n -1+2 n / n !) (C) (4 n +(-2) n / n !) (D) (4 n -1+(-2) n -1/ n !)

Tardigrade Admin · Accepted Answer

( e 7 x + e x / e 3 x )= e 4 x + e -2 x =[1+4 x +((4 x )2/2 !)+ ldots . .]+[1+(-2 x )+((-2 x )2/2 !)+ ldots ldots] ∴ coeff. of xn=(4n/n !)+((-2)n/n !)

Q. The co-efficient of $x^{n}$ in the expansion of $\frac{e ^{7 x} + e ^{x}}{e ^{3 x}}$ is

$\frac{4 ^{n - 1} + ( - 2 ) ^{n}}{n !}$

$\frac{4 ^{n - 1} + 2 ^{n}}{n !}$

$\frac{4 ^{n} + ( - 2 ) ^{n}}{n !}$

$\frac{4 ^{n - 1} + ( - 2 ) ^{n - 1}}{n !}$

$\frac{e ^{7 x} + e ^{x}}{e ^{3 x}} = e^{4 x} + e^{- 2 x}$
$= [1 + 4 x + \frac{( 4 x ) ^{2}}{2 !} + \dots ..] + [1 + (- 2 x) + \frac{( - 2 x ) ^{2}}{2 !} + \dots\dots]$
$∴$ coeff. of $x^{n} = \frac{4 ^{n}}{n !} + \frac{( - 2 ) ^{n}}{n !}$

Q. The co-efficient of xn in the expansion of e3xe7x+ex​ is

n!4n−1+(−2)n​

n!4n−1+2n​

n!4n+(−2)n​

n!4n−1+(−2)n−1​