First slide

Binomial theorem for positive integral Index

Question

The largest term in the expansion of $(3 + 2 x)^{50}$ where $x = \frac{1}{5}$ is

Moderate

A

5th
B

3th
C

7th
D

6th

Solution

$∵ (3 + 2 x)^{50} = 3^{50} {(1 + \frac{2 x}{3})}^{50}$
$Here, T_{r + 1} = {3^{50}}^{50} C_{r} {(\frac{2 x}{3})}^{r}$
$and T_{r} = 3^{5050} C_{r - 1} {(\frac{2 x}{3})}^{r - 1}$
$but x = \frac{1}{5} (given)$
$\begin{aligned} ∴ \frac{T_{r + 1}}{T_{r}} \geq 1 \Rightarrow \frac{^{50} C_{r}}{^{50} C_{r - 1}} \frac{2}{3} \cdot \frac{1}{5} \geq 1 \\ \Rightarrow 102 - 2 r \geq 15 r \Rightarrow r \leq 6 \end{aligned}$

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