Proving Primness in a summation

Question

I've been hitting my head against the wall for a little bit trying to figure out where to get started on proving (or disproving) the following:

$\exists k \in \mathbb{Z} $ such that$ \sum\limits_{i=0}^{|k|} i^2 + 2i $ is a prime number$ $

My first instinct tells me that it cannot be prime, due to the fact that for a number like 5, the result would be 35, which is not prime. Could it be I am mis-interpreting the $\exists$? do I only need to prove that a single prime number could exist under the conditions? thanks in advance guys!

Answer 1

$i*i+2i$

$i(i+2)$

$i=1$

$1(1+2)$

$1(3)$

$3$ #is a prime number

Therefore there exists some integer k(1) such that the statement is true

Answer 2

From the original question, you are confused about the things being summed and the sum itself. in particular, $35$ is never considered. The sum comes out $$ 0,3,11,26,50,85,133,196,\ldots $$ For a specific $k \geq 0, $ the sum is exactly $$ \frac{k(k+1)(2k+7)}{6}. $$ For $k > 6$ the thing is automatically composite, since the $6$ divides either into one of the factors or $2$ and $3$ do, leaving three factors larger than one. So the thing is prime only for $k=1,2.$

I imagine they are trying to emphasize that a single example is enough to show the truth of existence.