Q67 / 155 · WBJEE 2018 · Mathematics

If $n$ is even posirive integer, then the condition that the greatest term in the expansion of $(a+x)^{n}$ may also have the greatest coefficient, is

  • Correct answer
    A
    $\frac{n}{n+2} < x < \frac{n+2}{n}$
  • B
    $\frac{n}{n+1} < x < \frac{n+1}{n}$
  • C
    $\frac{n+1}{n+2} < x < \frac{n+2}{n+1}$
  • D
    $\frac{n+2}{n+3} < x < \frac{n+3}{n+2}$

Solution

For greatest term of $(1+x)^{n},$ we have $\quad \frac{n}{2} < \frac{n+1}{1+x} < \frac{n}{2}+1$ $\Rightarrow \quad \frac{n}{2} < \frac{n+1}{1+x}$ and $\frac{n+1}{1+x} < \frac{n}{2}+1$ $\Rightarrow \quad 1+x < \frac{n+1}{n / 2}$ and $\frac{n+1}{\frac{n}{2}+1} < 1+x$ $\Rightarrow \quad x < \frac{n+1-n / 2}{n / 2}$ and $\quad \frac{n+1-(n / 2-1)}{\frac{n+2}{2}} < x$ $\Rightarrow \quad x < \frac{n+2}{n}$ and $\frac{n}{n+2} < x$ $\Rightarrow \quad \frac{n}{n+2} < x < \frac{n+2}{n}$

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

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