For which positive integers n is the ratio, ( displaystyle∑ k =1 n k 2/ displaystyle∑ k =1 n k ) an integer?

Question

For which positive integers n is the ratio, ( displaystyle∑ k =1 n k 2/ displaystyle∑ k =1 n k ) an integer? (A) odd n only (B) even n only (C) n =1+6 k only, where k ≥ 0 and k ∈ I (D) n =1+3 k, integer k ≥ 0

Tardigrade Admin · Accepted Answer

( n ( n +1)(2 n +1) ⋅ 2/6 ⋅ n ( n +1)) must be an integer (2 n +1/3) must be an integer ⇒ (2 n +1) is divisible by 3 ⇒ n ∈ 1,4,7,10, ldots ldots ., n is of the form of (3 k +1), k ≥ 0, k ∈ I

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

odd $n$ only

even $n$ only

$n = 1 + 6 k$ only, where $k \geq 0$ and $k \in I$

$n = 1 + 3 k$ , integer $k \geq 0$

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

Q. For which positive integers n is the ratio, k=1∑n​kk=1∑n​k2​ an integer?

odd n only

even n only

n=1+6k only, where k≥0 and k∈I

n=1+3k, integer k≥0