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If the variance of the terms in an increasing arithmetic progression (AP) \((b_1, b_2, b_3, \ldots, b_{11})\) is 90, then what is the common difference of the AP?

A. 1
B. 3
C. 5
D. 9

Answer :

Final answer:

The common difference of an increasing arithmetic progression with a variance of 90 and 11 terms is 3.

Explanation:

To find the common difference of an arithmetic progression (AP) using the variance, we need to employ the formula for the variance of an AP. The variance, denoted as σ^2, for an AP with n terms is given by σ^2 = (n^2 - 1) × d^2 / 12, where d is the common difference and n is the number of terms. Given that the variance is 90 for 11 terms, we can substitute these values into the formula to solve for d.

So, 90 = (11^2 - 1) × d^2 / 12. Simplifying, we get 90 = 120 × d^2 / 12. Further simplifying gives us d^2 = 9. Taking the square root, we find that d can be ±3. However, since we are given that the AP is increasing, the common difference must be positive. Therefore, the common difference of the AP is 3.

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