If the variance of the terms of an increasing A.P.

$b_1,b_2,\dots ,b_{11}$ is 90, then the common difference of this A.P. is

Let $d$ be the common difference of the AP. then,

$\begin{array}{l}{b}_i=b_1+(i-1)di=1,2,\dots ,11.\\ \Rightarrow \mathrm{Var}\left(b_1,b_2,\dots ,b_{11}\right)=\mathrm{Var}(0d,1d,2d,\dots ,10d)\\ \Rightarrow 90=\mathrm{Var}(d,2d,3d,\dots ,10d)\\ \Rightarrow 90=d^2\mathrm{Var}(1,2,3,\dots ,10)\end{array}$

$\begin{array}{l}\Rightarrow 90=d^2\left(\frac{{10}^2-1}{12}\right)\\ \Rightarrow 90=10d^2\Rightarrow d=3\left[\begin{array}{c}\because b_1,b_2,\dots ,b_{11}\text{ is an increasing A.P. }\\ \therefore d>0\end{array}\right]\end{array}$