The variance of the series $a,a+d,a+2d,a+3d,....a+2nd$ is $\cfrac { n(n+1) }{ 3 } { d }^{ 2 }$.
-
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
-
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
-
Assertion is correct but Reason is incorrect
-
Assertion is incorrect but Reason is correct
To verify the assertion, the arithmetic mean of the series is calculated as a plus nd. The variance of the series is found by taking the average of the squared deviations from the mean, specifically the sum from k equals negative n to n of (kd) squared divided by (2n plus 1). Expanding the numerator gives d squared times n times (n+1) times (2n+1) divided by 3, and dividing this by the denominator (2n+1) confirms the variance is n(n+1) divided by 3 times d squared. Because the variance formula is derived directly from the algebraic identity for the sum of the squares of the first n integers, both the assertion and the reason are correct, and the reason perfectly explains the assertion.