Question

If the variance of the terms in an increasing $A.P., b _{1}, b _{2}, b _{3}, \ldots b _{11}$ is $90,$ then the common difference of this $A.P$ . is

Answer 1

Let a be the first term and $d$ be the common difference of the given $A.P$. Where $d >0$
$\bar{X}=a+\frac{0+d+2 d+\ldots+10 d}{11} $
$=a+5 d $
$\Rightarrow \text { varience }=\frac{\sum\left(\bar{X}-x_{i}\right)^{2}}{11} $
$\Rightarrow 90 \times 11=\left(25 d^{2}+16 d^{2}+9 d^{2}+4 d^{2}\right) \times 2 $
$\Rightarrow d=\pm 3 \Rightarrow d=3 $