First slide

Binomial theorem for positive integral Index

Question

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

Difficult

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

Here, the greatest coefficient is ²ⁿC_n.
$\begin{array}{ll} ∴^{2 n} C_{n} x^{n} >^{2 n} C_{n + 1} x^{n - 1} \Rightarrow x > \frac{n}{n + 1} \\ and^{2 n} C_{n} x^{n} >^{2 n} C_{n - 1} x^{n + 1} \Rightarrow x < \frac{n + 1}{n} \end{array}$
$∴ x mustlieintheinterval (\frac{n}{n + 1}, \frac{n + 1}{n})$

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