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

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: 3

Accepted Answer

Let common difference be $d$. Using $b_i = b_1+rd$ for $r=0,\ldots,10$:

$$\text{Variance} = \left(b_1^2+10b_1d+35d^2\right)-(b_1+5d)^2 = 10d^2$$

Given variance $=90$: $10d^2=90 \implies d=3$

Solutions · 1