Question

The interval in which x must lie so that the greatest term in the expression of $(1+x{)}^{2n}$ has the greatest coefficient, is

• A. $(\frac{n-1}{n},\frac{n}{n-1})$
• B. $(\frac{n}{n+1},\frac{n+1}{n})$
• C. $(\frac{n}{n+1},\frac{n+2}{n})$
• D. None of these

Answer 1

The correct option is B $(\frac{n}{n+1},\frac{n+1}{n})$

Here the greatest coefficient is ${}^{2n}{C}_n$
$\therefore {}^{2n}{C}_n{x}^n{>}^{2n}{C}_{n+1}{x}^{n-1}\Rightarrow x>\frac{n}{n+1}$
and ${}^{2n}{C}_n{x}^n{>}^{2n}{C}_{n-1}{x}^{n+1}\Rightarrow x<\frac{n+1}{n}$
Hence the required interval is $(\frac{n}{n+1},\frac{n+1}{n}).$