Question

If $n$ is even positive integer, then the condition that the greatest term in the expansion of ${(1+x)}^n$ may also have the greatest coefficient is

• 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. $\frac{n+2}{n+2}<x<\frac{n+3}{n+2}$

Answer 1

The correct option is A $\frac{n}{n+2}<x<\frac{n+2}{n}$

For greatest term we have
$\frac{n}{2}<\frac{n+1}{1+\left|x\right|}<\frac{n}{2}+1$
$\Rightarrow \frac{n}{2}<\frac{n+1}{1+x}$ and
$\frac{n+1}{1+\left|x\right|}\le \frac{n}{2}+1$
$\Rightarrow 1+x<\frac{n+1}{n/2}$ and
$\frac{n+1}{\frac{n}{2}+1}-1\le x$
$\Rightarrow x<\frac{n+1-\frac{n}{2}}{\frac{n}{2}}$ and
$\frac{n+1-\frac{n}{2}-1}{\frac{n+2}{2}}\le x$
$\Rightarrow x<\frac{n+2}{n}$ and
$\frac{n+2}{2}\le x$
$\therefore \frac{n}{n+2}<x<\frac{n+2}{n}$