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Q.
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
Binomial Theorem
Solution:
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 $x<\frac{n+2}{n}$
$\Rightarrow \frac{n}{n+2} < x < \frac{n+2}{n}$