If the coefficient of $r^{\text{th }},(r+1{)}^{\text{th }}$ and $(r+2{)}^{\text{th }}$terms in the expansion of $(1+x{)}^n$ are in A.P., then $n$ is a root of the equation 

${ }^n{C}_{r-1}{,}^n{C_{r\mathrm{\prime }}}^n{C}_{r+1}$

$\begin{array}{l}\Rightarrow 2\left({ }^n{C}_r\right){=}^n{C}_{r-1}{+}^n{C}_{r+1}\\ \Rightarrow 2=\frac{{ }^n{C}_{r-1}}{{ }^n{C}_r}+\frac{{ }^n{C}_r+1}{{ }^n{C}_r}=\frac{r}{n-r+1}+\frac{n-r}{r+1}\\ \Rightarrow 2(n-r+1)(r+1)=r^2+r+n^2-2nr+r^2+n-r\\ \Rightarrow n^2-(4r+1)n+4r^2-2=0\end{array}$