Q83 / 160 · WBJEE 2011 · Mathematics

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

  • A
    \(\frac{4^{\mathrm{n}-1}-(-2)^{\mathrm{n}-1}}{\lfloor\mathrm{n}}\)
  • B
    \(\frac{4^{n-1}-2^{n-1}}{\lfloor\text { n }}\)
  • C
    \(\frac{4^{\mathrm{n}}-2^{\mathrm{n}}}{\lfloor\mathrm{n}}\)
  • Correct answer
    D
    \(\frac{4^{\mathrm{n}}+(-2)^{\mathrm{n}}}{\underline{n}}\)

Solution

Hints: \(\frac{e^{7 x}+e^x}{e^{3 x}}=e^{4 x}+e^{-2 x}\) Co-efficient of \(x^n\) \(\frac{(4)^{\mathrm{n}}}{\mathrm{n} !}+\frac{(2)^{\mathrm{n}}}{\mathrm{n} !}(-1)^{\mathrm{n}}=\frac{4^{\mathrm{n}}+(-2)^{\mathrm{n}}}{\mathrm{n} !}\)

Hence, the correct answer is option: 'D'.

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