Induction proves a statement for every positive integer by linking one case to the next.
Base case n=1 gives 1=1. Assume the formula holds at n. Adding n+1 yields n(n+1)/2+(n+1)=(n+1)(n+2)/2, the required formula for the next case. The staircase rectangle supplies a separate visual explanation of the same sum.
Checking many examples is evidence, not a proof for all positive integers.