The coefficient of $\mathrm{r}^{ ext {th }}$ and $(\mathrm{r}+1)^{ ext {th }}$ terms in the expans

Question

The coefficient of $\mathrm{r}^{	ext {th }}$ and $(\mathrm{r}+1)^{	ext {th }}$ terms in the expansion of $(1+\mathrm{x})^{20}$ are in the ratio $1: 2$, then $\mathrm{r}=$

Professor PC Thomas Classes · Answer

$\mathrm{T}_{\mathrm{r}+1}={ }^{20} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{\mathrm{r}} 	herefore$ coefficient of $ \  \begin{aligned} & \mathrm{T}_{\mathrm{r}+1}={ }^{20} \mathrm{C}_{\mathrm{r}} 	herefore \frac{{ }^{20} \mathrm{C}_{\mathrm{r}-1}}{{ }^{20} \mathrm{C}_{\mathrm{r}}}=\frac{1}{2} \ & \Rightarrow \frac{20!}{(\mathrm{r}-1)!(21-\mathrm{r}!)} 	imes \frac{\mathrm{r}!(20-\mathrm{r})!}{20!}=\frac{1}{2} \ & \Rightarrow \frac{\mathrm{r}}{21-\mathrm{r}}=\frac{1}{2} \Rightarrow \mathrm{r}=7 \end{aligned} $