Question

The two consecutive terms in the expansion of $(3+2x{)}^{74}$ whose coefficients are equal, are

• A. 11, 12
• B. 7, 8
• C. 30, 31
• D. None of these

Answer 1

Answer: (C)

General term of $(3+2x{)}^{74}$ is
$T_{r+1}={}^{74}{C}_r(3{)}^{74-r}{2}^r{x}^r$
Let two consecutive terms are $T_{r+1}th$ and $T_{r+2}th$ terms
According to the given condition,
Coefficient of $T_{r+1}=$ Coefficient of $T_{r+2}$
${\Rightarrow }^{74}{C}_r{3}^{74-r}{2}^r={}^{74}{C}_{r+1}{3}^{74-(r+1)}{2}^{r+1}$
$\frac{{}^{74}{C}_{r+1}}{{}^{74}{C}_r}=\frac{3}{2}$
$\Rightarrow \frac{74-r}{r+1}=\frac{3}{2}$
$\Rightarrow 148-2r=3r+3$
$\Rightarrow r=29$
Hence, two consecutive terms are $30$ and $31$.