TITLE: Intuitively understanding $\sum_{i=1}^ni={n+1\choose2}$
QUESTION [10 upvotes]: It's straightforward to show that
$$\sum_{i=1}^ni=\frac{n(n+1)}{2}={n+1\choose2}$$
but intuitively, this is hard to grasp. Should I understand this to be coincidence? Why does the sum of the first $n$ natural numbers count the number of ways I can choose a pair out of $n+1$ objects? What's the intuition behind this?

REPLY [3 votes]: This is the classic proof without words, from https://maybemath.wordpress.com/

That doesn't help with this part of your question:

Why does the sum of the first $n$ natural numbers count the number of
  ways I can choose a pair out of $n+1$ objects? 

Here's a way to rephrase @user17762 's excellent accepted answer.
Imagine $n+1$ kids in a room. Each shakes hands with all the others. Then each kid shakes hands $n$ times, so there are $n(n+1)$ handshakes - each counted twice. You can pick a pair of kids (that is, a handshake) in $n(n+1)/2$ ways. But you can also think about the kids shaking hands as they enter the room one at a time. The second kid coming has one hand to shake. The third has two, and so on, for a total of $1 + 2 + \cdots + n$.