Two successive terms in the expansion of ${(1+x)}^{24}$ whose coefficients are in the ratio $4:1$ are 

Let $r^{\text{th}},{\left(r\text{+1}\right)}^{\text{th}}$ terms are two successive terms

$t_r={}^{24}{\text{C}}_{r-1}$

$t_{r+1}={}^{24}{\text{C}}_r$

$\frac{t_r}{t_{r+1}}=\frac{{ }^{24}{\text{C}}_{r-1}}{{ }^{24}{\text{C}}_r}=\frac{4}{1}$

$\frac{r}{24-r+1}=\frac{4}{1}$

$r=100-4r$

$r=\frac{100}{5}=20$

$\therefore {20}^{\text{th}},{21}^{st}$ terms coefficients are in the ratio $4:1$