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Question

# Assertion :The variance of first n even natural numbers is n2âˆ’14. 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 correctSum 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(n−1)=n2−13

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