Question

If the coefficients of $r^{th}$ and $(r+1{)}^{th}$ terms in the expansion of $(3+7x{)}^{29}$ are equal, then $r=$

• A. 14
• B. 15
• C. 18
• D. 21

Answer 1

The correct option is D 21

The $r$th term in the expansion of ${(3+7x)}^{29}$ is ${{}^{29}C}_{r-1}{\left(3\right)}^{30-r}{\left(7x\right)}^{r-1}$

The coefficient of $r$th term is ${{}^{29}C}_{r-1}{\left(3\right)}^{30-r}{\left(7\right)}^{r-1}$

Similarly the coeficient of $(r+1)$th term is ${{}^{29}C}_r{\left(3\right)}^{29-r}{\left(7\right)}^r$

Given that both coefficients are equal , so we get

${{}^{29}C}_r{\left(3\right)}^{29-r}{\left(7\right)}^r={{}^{29}C}_{r-1}{\left(3\right)}^{30-r}{\left(7\right)}^{r-1}$

$\frac{29-r+1}{r}=\frac{3}{7}$

$\Rightarrow \frac{7}{r}=\frac{3}{30-r}\Rightarrow 210-7r=3r\Rightarrow 210=10r$

$\Rightarrow r=21$

Therefore the correct option is $D$