First slide

Binomial theorem for positive integral Index

Question

The two consecutive terms whose coefficients in the expanssion of ${(3 + 2 x)}^{94}$ are equal, are

Moderate

A

$37 t h, 38 t h$
B

$38 t h, 39 t h$
C

$47 t h, 48 t h$
D

$48 t h, 49 t h$

Solution

Suppose coefficients of $(r + 1)$ th and $(r + 2) t h$ terms in the expansion of ${(3 + 2 x)}^{94}$ are equal. Then
$^{94} C_{r} 3^{94 - r} 2^{r} =^{94} C_{r + 1} 3^{94 - (r + 1)} 2^{r + 1}$
$\begin{array}{l} \Rightarrow \frac{94!}{r! (94 - r)!} \cdot \frac{(r + 1)! (94 - r - 1)!}{94!} = \frac{2}{3} \\ \Rightarrow \frac{r + 1}{94 - r} = \frac{2}{3} \Rightarrow 3 r + 3 = 188 - 2 r \\ \Rightarrow 5 r = 185 \Rightarrow r = 37 \end{array}$

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