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BITSAT2024MathematicsBinomial TheoremActual

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

Options

  1. A4^ n-1 +(-2)^n n!
  2. B4^ n-1 +2^n n!
  3. C4^n+(-2)^n n!
  4. D4^ n-1 +(-2)^ n-1 n!

Correct answer

C. 4^n+(-2)^n n!

Step-by-step solution

Given, e^ 7 x +e^x e^ 3 x e^ 4 x +e^ -2 x Series of e^a=1+ a 1! + a^2 2! + a^3 3! aligned e^ 4 x + & e^ -2 x = (1+ 4 x 1! + (4 x)^2 2! + (4 x)^3 3! + ) & + (1+ (-2 x) 1! + (-2 x)^2 2! + ) aligned Here, coefficient of aligned & x^2 4^2 2! + (-2)^2 2! & x^3 4^3 3! + (-2)^3 3! & & x^n= 4^n n! + (-2)^n n! aligned

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