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Question

Assertion :The variance of first n even natural numbers is n214. Reason: The sum of first n natural even numbers is n(n+1) and the sum of squares of first n natural numbers is n(n+1)(2n+1)6

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Assertion is incorrect but Reason is correct
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Solution

The correct option is D Assertion is incorrect but Reason is correct
Sum of first n even natural numbers
=2+4+6+...+2n=2(1+2+...+n)
=2n(n+1)2=n(n+1)
Mean (¯x)=n(n+1)n=n+1
variance =1n(x1)2(¯x)2=1n(22+42+...+(2n)2)(n+1)2
=1n22(12+22+...+n2)(n+1)2
=4nn(n+1)(2n+1)6(n+1)2
=23(n+1)(2n+1)(n+1)2
=n+13[2(2n+1)3(n+1)]
=(n+1)3(n1)=n213

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