In logic and mathematics, an argument that establishes a proposition's validity. Formally, it is a finite sequence of formulas generated according to accepted rules. Each formula either is an axiom or is derived from a previously established theorem, and the last formula is the statement that is to be proven. The essence of deductive reasoning (see deduction), this is the basis of Euclidean geometry and all scientific methods inspired by it. An alternative form of proof, called mathematical induction, applies to propositions defined through processes based on the counting numbers. If the proposition holds for math.n = 1 and can be shown to hold for math.n = math.k + 1 whenever math.n = math.k (a constant) is also true, then it holds for all values of math.n. An example is the assertion that the sum of the first math.n counting numbers is math.n(math.n + 1)/2.

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