If the variance of the terms in an increasing

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________.

Solution:

Variance $=\frac{\sum_{i=1}^{11} b_{i}^{2}}{11}-\left(\frac{\sum_{i=1}^{11} b_{i}}{11}\right)^{2}$

Let common difference of A.P. be $d$

$=\frac{\sum_{r=0}^{10}\left(b_{1}+r d\right)^{2}}{11}-\left(\frac{\sum_{r=0}^{10}\left(b_{1}+r d\right)}{11}\right)^{2}$

$=\frac{11 b_{1}^{2}+2 b_{1} d\left(\frac{10 \times 11}{2}\right)+d^{2}\left(\frac{10 \times 11 \times 21}{6}\right)}{11}$

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