The coefficient of x n in the expansion of e 7 x + e x e 3 x is

Explanation: e 7 x + e π e 3 x = e 4 x + e − 2 x Co-efficient of x n ( 4 ) n n ! + ( 2 ) n n ! ( − 1 ) n = 4 n + ( − 2 ) n n !

(A) 4 n − 1 − ( − 2 ) n − 1 ⌊ n

(B) 4 n − 1 − 2 n − 1 ⌊ n

The coefficient of x n in the expansion of e 7 x + e x e 3 x is

(A) 4 n − 1 − ( − 2 ) n − 1 ⌊ n

(B) 4 n − 1 − 2 n − 1 ⌊ n

Question:

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

WBJEE

Updated On: Aug 1, 2023

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

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The Correct Option is D

Solution and Explanation

Explanation:

\frac{e^{7 x} + e^{π}}{e^{3 x}} = e^{4 x} + e^{- 2 x}

Co-efficient of

x^{n}

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

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