First slide

Binomial theorem for positive integral Index

Question

For which of the following values of x, 5th term is the numerically greatest term in the expansion of $(1 + x / 3)^{10}$

Moderate

A

-2
B

1.8
C

2
D

-1.9

Solution

Let T₅ be numerically the greatest term in the expansion of
$\begin{array}{l} Then, |\frac{T_{5}}{T_{4}}| \geq 1 and |\frac{T_{6}}{T_{5}}| \leq 1 \\ Now, \frac{T_{r + 1}}{T_{r}} = \frac{10 - r + 1}{r} \frac{x}{3} \\ \Rightarrow |\frac{7}{4} \times \frac{x}{3}| \geq 1 and |\frac{6}{5} \times \frac{x}{3}| \leq 1 \\ \Rightarrow | x | \geq \frac{12}{7} and | x | \leq \frac{5}{2} \\ \Rightarrow \frac{12}{7} \leq | x | \leq \frac{5}{2} \\ \Rightarrow x \in [- \frac{5}{2}, - \frac{12}{7}] \cup [\frac{12}{7}, \frac{5}{2}] \end{array}$

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