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

Real Analysis · Axiom Academy

EXAMPLE Proving the Sum Formula by Induction Work through a complete proof that for every positive integer. Prove that the sum of the first n positive integers is given by the closed form below, using the Principle of Mathematical Induction . You just completed a full proof by mathematical induction. Every such proof has the same four moving parts: Base case: verify the statement for the smallest value (here n = 1 ). Inductive hypothesis: assume the statement holds for an arbitrary fixed k . Inductive step: use that assumption to prove the statement for k+1 — the whole argument turns on substituting the hypothesis. Conclusion: base case + step the statement holds for all . Induction is the workhorse for proving statements indexed by — sums, inequalities, divisibility, recursively defined objects. The same skeleton you used here transfers directly to all of them.

This is the written version of the interactive lesson above. See the full Real Analysis course.