Proving Sum Formula 1+2+...+n = n(n+1)/2

Intro to Proofs · Axiom Academy

EXAMPLE Proving the Sum Formula by Induction The classic proof of 1 + 2 + ⋯ + n = n(n+1)/2, worked step by step with mathematical induction. Show that the sum of the first n positive integers is given by a closed formula — for every positive integer n , not just a few test cases. We prove it by mathematical induction. Excellent work — you have completed a full proof by mathematical induction. Here is what to carry forward: The base case is essential: always verify the formula at the smallest value (here n = 1 ). Without it, the chain has nothing to start from. The inductive hypothesis is a tool, not a cheat: assuming the statement holds for an arbitrary k is exactly what lets you prove it for k+1 . The inductive step does the work: use the hypothesis to climb from k to k+1 . True for 1 forces true for 2 , then 3 , and so on — the domino effect. The algebra is the craft: the key move was splitting off (k+1) , substituting the hypothesis, then factoring (k+1) out to reach the target. The structure repeats: base case, hypothesis, inductive step — every induction proof follows this same three-part shape. Induction is everywhere: you will use it to prove divisibility results, inequalities, and recursion formulas throughout your studies.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.