Question

For which positive integers $n$ is the ratio, $\frac{\displaystyle\sum_{ k =1}^{ n } k ^2}{\displaystyle\sum_{ k =1}^{ n } k }$ an integer?

• A. odd $n$ only
• B. even $n$ only
• C. $n =1+6 k$ only, where $k \geq 0$ and $k \in I$
• D. $n =1+3 k$, integer $k \geq 0$

Answer 1

Answer: (D)

$\frac{ n ( n +1)(2 n +1) \cdot 2}{6 \cdot n ( n +1)}$ must be an integer
$\frac{2 n +1}{3}$ must be an integer $ \Rightarrow (2 n +1)$ is divisible by 3
$\Rightarrow n \in 1,4,7,10, \ldots \ldots ., n$ is of the form of $(3 k +1), k \geq 0, k \in I$