Question

If the mean of the squares of first $n$ natural numbers is $105$, then the median of first $n$ natural numbers is

• A. $8$
• B. $9$
• C. $10$
• D. $11$

Answer 1

Answer: (B)

$9$

We know mean of the squares of first $n$ natural number is,

$=\frac{1^2+2^2+...+n^2}{n}=105$ ...(Given)

$\Rightarrow \frac{(n+1)(2n+1)}{6}=105$

$\Rightarrow 2n^2+3n-629=0$

$\Rightarrow (2n+37)(n-17)=0$

$\Rightarrow n=17$

Thus, given numbers are $1,2,3,4,...,16,17$.

Clearly, $9$ is their median.