Question

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

• A. $\frac{n}{n+2}<x<\frac{n+2}{n}$
• B. $\frac{n+1}{n}<x<\frac{n}{n+1}$
• C. $\frac{n}{n+4}<x<\frac{n+4}{4}$
• D. None of these

Answer 1

Answer: (A)

If $n$ is even, then ${}^n{C}_{n/2}$ is the greatest coefficients in the expansion of $(1+x{)}^n$ and it occurs in ${\left(\frac{n}{2}+1\right)}^{\text{th }}$ term
For ${\left(\frac{n}{2}+1\right)}^{\text{th }}$ term to be the greatest term, we must have
$T_{\frac{n}{2}+1}>T_{\frac{n}{2}}$ and $T_{\frac{n}{2}+1}>T_{\frac{n}{2}+2}$
$\Rightarrow \frac{T_{\frac{n}{2}+1}}{T_{\frac{n}{2}}}>1$
and $\frac{T_{\frac{n}{2}+2}^2}{T_{\frac{n}{2}+1}}<1$
$\Rightarrow \frac{{}^n{C}_{\frac{n}{2}}}{{}^n{C}_{\frac{n}{2}1}}x>1$ and $\frac{{}^n{C}_{\frac{n}{2}+1}}{{}^n{C}_{\frac{n}{2}}}x<1$
$\Rightarrow \frac{n+2}{n}\cdot x>1$ and $\frac{n}{n+2}x<1$
$\Rightarrow x>\frac{n}{n+2}$ and $x<\frac{n+2}{n}$
$\Rightarrow \frac{n}{n+2}<x<\frac{n+2}{n}$