If the variance of the first n

Question:

If the variance of the first $n$ natural numbers is 10 and the variance of the first $m$ even natural numbers is 16 , then $m+n$ is equal to__________.

Solution:

$\operatorname{Var}(1,2, \ldots . ., n)=10$

$\Rightarrow \frac{1^{2}+2^{2}+\ldots . .+n^{2}}{n}-\left(\frac{1+2+\ldots . .+n}{n}\right)^{2}=10$

$\Rightarrow \frac{(n+1)(2 n+1)}{6}-\left(\frac{n+1}{2}\right)^{2}=10$

$\Rightarrow n^{2}-1=120 \Rightarrow n=11$

$\operatorname{Var}(2,4,6, \ldots . ., 2 m)=16 \Rightarrow \operatorname{Var}(1,2, \ldots . ., m)=4$

$\Rightarrow m^{2}-1=48 \Rightarrow m=7$

$\Rightarrow m+n=18$

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