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Q.
The two consecutive terms in the expansion of $(3+2 x)^{74}$ whose coefficients are equal, are
ManipalManipal 2008
Solution:
General term of $(3+2 x)^{74}$ is
$T_{r+1}={ }^{74} C_{r}(3)^{74-r} 2^{r} x^{r}$
Let two consecutive terms are $T_{r+1} t h$ and $T_{r+2} t h$ terms
According to the given condition,
Coefficient of $T_{r+1}=$ Coefficient of $T_{r+2}$
$\Rightarrow ^{74} C_{r} 3^{74-r} 2^{r}={ }^{74} C_{r+1} 3^{74-(r+1)} 2^{r+1}$
$\frac{{ }^{74} C_{r+1}}{{ }^{74} C_{r}}=\frac{3}{2}$
$\Rightarrow \frac{74-r}{r+1}=\frac{3}{2}$
$\Rightarrow 148-2 r=3 r+3$
$\Rightarrow r=29$
Hence, two consecutive terms are $30$ and $31$.