First slide

Binomial theorem for positive integral Index

Question

The interval in which $x (> 0)$ must lie so that the greatest term in the expansion of
${(1 + x)}^{2 n}$ has the greatest coefficient is

Moderate

A

$(\frac{n - 1}{n}, \frac{n}{n - 1})$
B

$(\frac{n}{n + 1}, \frac{n + 1}{n})$
C

$(\frac{n}{n + 2}, \frac{n + 2}{n})$
D

none of these

Solution

The greatest coefficient in the expansion of $(1 + x)^{2 n}$ is $C_{n}$ . We are given $^{2 n} C_{n} x^{n}$ is the greatest term.
$∴^{2 n} C_{n - 1} x^{n - 1} <^{2 n} C_{n} x^{n}$
$^{2 n} C_{n + 1} x^{n + 1} <^{2 n} C_{n} x^{n}$
$\begin{array}{l} \Rightarrow \frac{^{2 n} C_{n - 1}}{^{2 n} C_{n}} < x < \frac{^{2 n} C_{n}}{^{2 n} C_{n + 1}} \\ \Rightarrow \frac{n}{n + 1} < x < \frac{n + 1}{n} . \end{array}$

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