math / Foundations / free
Proofs & Reasoning
See how definitions and small steps support a general claim.
Follow the mechanism1 / 3
A claim needs a precise target. A proof starts by naming exactly what must be shown, so examples and assumptions cannot quietly change the question.
proof by inductionP(1) → P(1)
Focus the idea
Claim for every positive integer n1 + 3 + 5 + ⋯ + (2n − 1) = n²
Inductive stepS(k + 1) = S(k) + (2k + 1) = k² + 2k + 1 = (k + 1)²
base caseproven
through n1
next odd term3
claim statuslinked
The base case is secure, and the ladder currently reaches n = 1. Prove the link from P(1) to P(2) to keep the chain moving.
Induction is not checking a long list of examples. It proves one base case and one universal link: whenever P(k) is true, P(k + 1) must be true. Together, those two statements cover the whole chain.