mathematical induction
Mathematical induction is a proof method for showing that a statement holds for every natural number — for n = 1, 2, 3, and on forever. The picture is a line of dominoes: knock over the first, and arrange them so each one topples the next, and you can be sure every domino falls.
A proof has two steps. The base case: check the statement is true for the starting value (usually n = 1). The inductive step: assume the statement is true for some n = k (the inductive hypothesis), and use that to prove it for n = k + 1. Together these guarantee it for all n, because the truth at one value relentlessly propagates to the next.
Both steps are essential. Skip the base case and the chain has nothing to start it; skip the inductive step and there is no guarantee the truth carries forward. Induction proves a statement is always true, which is far stronger than checking a few cases — a pattern that holds for n = 1 through 40 can still fail at 41.
To prove 1 + 2 + ... + n = n(n+1)/2: base n = 1 gives 1 = 1·2/2 ✓; assuming it for k, adding (k+1) yields k(k+1)/2 + (k+1) = (k+1)(k+2)/2, the formula for k+1.
Base case plus inductive step proves the sum formula for all n.