Question

If the variance of first $n$ even natural numbers is $133$ , then the value of $n$ is equal to

• A. $19$
• B. $24$
• C. $21$
• D. $20$

Answer 1

Answer: (D)

$Var\left(2,4 , 6 . . . . . . . . . 2 n\right)=133$
$\Rightarrow 4Var\left(1,2 , 3 . . . . . . . . . n\right)=133$
$\Rightarrow Var\left(1,2 , 3 . . . . . . . . . n\right)=\frac{133}{4}$
$\Rightarrow \frac{1^{2} + 2^{2} + . . . . . . . n^{2}}{n}-\left(\frac{1 + 2 + 3 + . . . . . . . n}{n}\right)^{2}=\frac{133}{4}$
$\Rightarrow \frac{\left(n + 1\right) \left(2 n + 1\right)}{6}-\frac{\left(n + 1\right)^{2}}{4}=\frac{133}{4}$
$\Rightarrow \frac{\left(n + 1\right)}{2}\left[\frac{2 n + 1}{3} - \frac{n + 1}{2}\right]=\frac{133}{4}$
$\Rightarrow \frac{\left(n + 1\right)}{2}\left[\frac{n - 1}{6}\right]=\frac{133}{4}$
$\Rightarrow n^{2}-1=399\Rightarrow n=20$