If the coefficients of the  ${(r + 2)}^{th}$ and  $r^{th}, {(r + 1)}^{th}$terms in the binomial expansion of ${(1 + y)}^m$ are in , then  $A.P.$$m$ and $r$ satisfy the equation 

  Coefficient of the $r\text{ th }$ term is ${ }^m{C}_{r-1}$

According to the given condition ${\mathrm{n}}^m{C}_{r-1}{+}^m{C}_{r+1}=2\left({ }^m{C}_r\right)$

$\begin{array}{l}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\frac{{ }^m{C}_{r-1}}{{ }^m{C}_r}+\frac{{ }^m{C}_{r+1}}{{ }^m{C}_r}=2\Rightarrow \frac{r}{m+1-r}+\frac{m-r}{r+1}=2\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{r}^2+r+(m-r)(m+1-r)=2(m+1-r)(r+1)\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{r}^2+r+m^2+(1-2r)m-r(1-r)=\\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\Rightarrow 2m(r+1)+2\left(1-r^2\right)\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{m}^2-m(4r+1)+4r^2-2=0.\end{array}$