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What is mathematical induction? Mathematical induction is a powerful method used to establish that a given statement (i.e. a theorem or a formula) is TRUE for all natural numbers. Note: The principle of mathematical induction is in fact a fully rigorous and deductive form of reasoning, not to be confused with inductive reasoning. The method of a proof by mathematical induction goes as follows: Let S (n ) be a statement which holds for any natural n (i.e. n 1, 2,3,... ) If 1) S (1) is TRUE, and 2) S (k 1) is TRUE whenever S ( k ) is TRUE, then the statement S (n ) is TRUE for all n {1,2,3,...} . The first step is called the base case and the second step is called the inductive step. The assumption in the inductive step that S ( k ) is true is called the induction hypothesis. Example Below, we prove that the sum of the first n natural numbers is given by the formula S ( n) 1 2 3 n 1) n (n 1) . 2 Base Case S (1) 1 1 2 . 2 The formula is (trivially) TRUE for n 1. 2) Inductive Step First we assume that S ( k ) is TRUE. We then have S (k ) 1 2 3 k k (k 1) . 2 This is the induction hypothesis. Now we use this hypothesis to prove that S (k 1) is TRUE, or that S (k 1) (k 1) (k 2) . 2 Adding an additional term to the sum of the first k natural numbers, namely k 1 , yields: S (k 1) 1 2 3 k (k 1) S (k ) (k 1) k (k 1) (k 1) 2 k (k 1) 2(k 1) 2 (k 1) ( k 2) 2 This finalizes the proof by induction.