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

$T_{r+1}={29}_{C_r}{\left(3\right)}^{29-r}{\left(7x\right)}^r$

$T_r={29}_{C_{r-1}}{\left(3\right)}^{30-r}{{\left(7x\right)}^r}^{-1}$

Given coeff of  $Tr=$Coeff of  $T_{r+1}$

$\Rightarrow {29}_{c_{r-1}}{3}^{30-r}{.7}^{r-1}={29}_{c_r}{3}^{29-r}{7}^r$

$\Rightarrow 3=\frac{{29}_{C_r}}{{29}_{C_{r-1}}}.7\Rightarrow 3=\frac{29-r+1}{r}.7$

$\Rightarrow 10r=210$

$\therefore r=21$