First slide

Binomial theorem for positive integral Index

Question

Numerically the greatest term in the expansion of ${(2 + 3 x)}^{9} when x = \frac{3}{2}, is$

Moderate

A

$\frac{7 \times 3^{3}}{2}$
B

$\frac{7 \times 3^{13}}{2}$
C

$\frac{3^{13}}{2}$
D

None of these

Solution

since $(2 + 3 x)^{9} = 2^{9} {(1 + \frac{3 x}{2})}^{9}$
now in the expansion of ${(1 + \frac{3}{2} x)}^{9}$ , we have
$\begin{aligned} \frac{T_{r + 1}}{T_{r}} = (\frac{9 - r + 1}{r}) |\frac{3}{2} x| \\ = (\frac{10 - r}{r}) (\frac{9}{4}) = \frac{90 - 9 r}{4 r} (∵ x = \frac{3}{2}) \\ ∵ \frac{T_{r + 1}}{T_{t}} \geq 1 \Rightarrow \frac{90 - 9 r}{4 r} \geq 1 \Rightarrow 90 \geq 13 r \\ \Rightarrow r \leq \frac{90}{13} \Rightarrow r \leq 6 \frac{12}{13} \end{aligned}$
.'. Maximum value of r is 6.

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