Binomial theorem for positive integral Index

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Question

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

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Solution

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

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Binomial theorem for positive integral Index

Remember concepts with our Masterclasses.

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

The greatest coefficient in the expansion of (1+x)2n is 2nCn We are given 2nCnxn is the greatest term ∴ 2nCn−1xn−1<2nCnxn 2nCn+1xn+1<2nCnxn⇒ 2nCn−1 2nCn<x< 2nCn 2nCn+1⇒nn+1<x<n+1n.

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