Assertion (A): The variance of the first n odd natural numbers is $\frac{n^{2} - 1}{3} .$
Reason (R) : The sum of the first n odd natural numbers is n² and the sum of the squares of the first n odd natural numbers is $\frac{n (4 n^{2} - 1)}{3} .$

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(A) is false, but (R) is true

(A) and (R) are true. (R) is correct explanation of (A)

(A) is true, but (R) is false

(A) and (R) are true, but (R) is not a correct explanation of (A)

answer is A.

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Detailed Solution

First n natural numbers 1.2.3........n

$Here \sum x_{i} = \frac{n (n + 1)}{2}, \sum x_{i}^{2} = \frac{n (n + 1) (2 n + 1)}{6}$

$variance is σ^{2} = \frac{\sum x_{i}^{2}}{n} - {(\frac{\sum x_{i}}{n})}^{2} = \frac{n^{2} - 1}{12}$

$Clearly Assertion is correct, obviously Reason is correct.$

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