If the variance of the terms in an increasing A.P.,
b₁ , b₂ , b₃ ,....,b₁₁ is 90, then the common difference of this A.P. is_______.

Let the common difference = d<br><br> and ${b_1} = a$<br> ${b_2} = a + d$<br> ${b_3} = a + 2d$ <br> ... ${b_{11}} = a + 10d$<br><br> Variance = $${{\sum {a_i^2} } \over {11}} - {\left( {{{\sum {{a_i}} } \over {11}}} \right)^2} = 90$$<br><br> $$ \Rightarrow {{{a^2} + {{\left( {a + d} \right)}^2} + ... + {{\left( {a + 10d} \right)}^2}} \over {11}} - {\left( {{{a + \left( {a + d} \right) + ... + \left( {a + 10d} \right)} \over {11}}} \right)^2} = 90$$<br><br> $$ \Rightarrow 11\left[ {11{a^2} + 385{d^2} + 110ad} \right] - {\left[ {11a + 55d} \right]^2} = 10890$$<br><br> $\Rightarrow 1210{d^2} = 10890$<br><br> $\Rightarrow {d^2} = 9$<br><br> $\Rightarrow d = \pm 3$<br><br> As A.P is increasing so d should be positive<br><br> $\therefore$ d = 3