To prove that "1+2+⋯+n = n(n+1)/2" holds for every natural number n, you cannot check them one by one because there are infinitely many natural numbers. Instead, it is enough to show just two things — ① the first domino (n=1) falls, and ② whenever one domino falls, the very next domino must also fall. Once these two facts are established, all the dominoes will fall even though the chain continues forever.
Why are these two steps enough? In ①, we confirmed that the domino for n=1 (the first domino) falls. In ②→③, we confirmed that "if the kth domino falls, the (k+1)th domino must also fall."
Then, because domino 1 falls, domino 2 falls; because domino 2 falls, domino 3 falls; and so on… Since this chain continues without breaking, we can conclude that the statement holds for every natural number n. Move the k slider and check that the same pattern repeats no matter which point in the chain you examine.