Intuitively understanding $\sum_{i=1}^ni={n+1\choose2}$ 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?

The intuition is that for the pairs can be listed in the following way.
$$\begin{array}{ccccccc}
1,2 & & & & & & \\
1,3 & 2,3 & & & & & \\
1,4 & 2,4 & 3,4 & & & & \\
1,5 & 2,5 & 3,5 & 4,5 & & & \\
1,6 & 2,6 & 3,6 & 4,6 & 5,6 & & \\
1,\vdots & 2,\vdots & 3,\vdots & 4,\vdots & 5,\vdots &\ddots & \\
1,n+1 & 2,n+1 & 3,n+1 & 4,n+1 & 5,n+1 & \cdots & n,n+1 \\
\end{array}$$
Notice that each row has length $i$ for $i=1,\ldots,n$ since the number of pairs with maximum element $i+1$ is $i$. Therefore the total number of pairs, which is $\binom{n+1}{2}$ is $\displaystyle \sum_{i=1}^n i$.
