The interval in which x must lie so that the | vedclass

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(B) The greatest coefficient in the expansion of $(1 + x)^{2n}$ is the middle term coefficient,which is $^{2n}C_n$.For the term $T_{n+1} = ^{2n}C_n x^

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$\left( \frac{n}{n + 1}, \frac{n + 1}{n} \right)$

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(B) The greatest coefficient in the expansion of $(1 + x)^{2n}$ is the middle term coefficient,which is $^{2n}C_n$.For the term $T_{n+1} = ^{2n}C_n x^n$ to be the greatest term,it must satisfy $T_{n+1} \ge T_n$ and $T_{n+1} \ge T_{n+2}$.First,$T_{n+1} \ge T_n \Rightarrow ^{2n}C_n x^n \ge ^{2n}C_{n-1} x^{n-1}$.This simplifies to $x \ge \frac{^{2n}C_{n-1}}{^{2n}C_n} = \frac{n}{n+1}$.Second,$T_{n+1} \ge T_{n+2} \Rightarrow ^{2n}C_n x^n \ge ^{2n}C_{n+1} x^{n+1}$.This simplifies to $x \le \frac{^{2n}C_n}{^{2n}C_{n+1}} = \frac{n+1}{n}$.Thus,the required interval is $\left( \frac{n}{n + 1}, \frac{n + 1}{n} \right)$.

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

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