Question

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

• A. $16$
• B. $17$
• C. $15$
• D. $18$

Answer 1

The correct option is B $17$

Sum of squares of first $n$ natural numbers is $=\frac{n(n+1)(2n+1)}{6}$

Mean $=\frac{\text{sum}}{n}$

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

Given mean is $105$.

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

$\Rightarrow 630=(n+1)(2n+1)$

$\Rightarrow (17+1)\left(2\right(17)+1)=(n+1)(2n+1)$

$\Rightarrow n=17$