If the variance of the terms in an increasing A . P . b _ 1 b _ 2 ,...

Given${b}_{1}, {b}_{2}, {b}_{3}, \ldots . . {b}_{11},$ are in A . P .

$∴$ Variance of \left({b}_{1}, {b}_{2}, \ldots . . {b}_{11}\right) =$Variance of$( 0, d, 2d, \ldots . . 10d ) = 90$</p><p>$⇒ \dfrac{{∑}_{i = 1}^{11}{b}_{i}^{2}}{11} - {\left(\dfrac{{∑}_{i = 1}^{11}{b}_{i}}{11}\right)}^{2} = 90$</p><p>$⇒ \dfrac{{d}^{2}}{11} \cdot \dfrac{10 . 11 . 21}{6} - {d}^{2} \cdot \dfrac{55}{11} \cdot \dfrac{55}{11} = 90$</p><p>$⇒ 35{d}^{2} - 25{d}^{2} = 90$</p><p>$⇒ {d}^{2} = 9 ⇒ d = \pm 3$</p><p>It is an increasing A . P . ⇒ d$ is positive.

$⇒ d = 3$