Question

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

Answer 1

Given : $(3+7x{)}^{29}$
General term is
$T_{r+1}={}^{29}{C}_r\left(3{)}^{29-r}\right(7x{)}^r$
Coefficient of $(r+1{)}^{th}$ term
$={}^{29}{C}_r\times 3^{29-r}\times (7{)}^r$
Coefficient of $r^{th}$ term
$={}^{29}{C}_{r-1}\times 3^{30-r}\times (7{)}^{r-1}$

According to question,
${}^{29}{C}_r\times 3^{29-r}\times (7{)}^r={}^{29}{C}_{r-1}\times 3^{30-r}\times (7{)}^{r-1}\Rightarrow \frac{{}^{29}{C}_r}{{}^{29}{C}_{r-1}}=\frac{3}{7}\Rightarrow \frac{30-r}{r}=\frac{3}{7}\therefore r=21$