explain mathametical induction formula
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Mathematical induction can be used to prove that the following statement, P(n), holds for all natural numbers n. P(n) gives a formula for the sum of the natural numbers less than or equal to number n. ... Base case: Show that the statement holds for n = 0.
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The simplest and most common form of mathematical induction infers that a statement involving a natural number n holds for all values of n. The proof consists of two steps:
The base case: prove that the statement holds for the first natural number n. Usually, n= 0 or n = 1; rarely, but sometimes conveniently, the base value of n may be taken as a larger number, or even as a negative number (the statement only holds at and above that threshold), because these extensions do not disturb the property of being a well-ordered set).The step case or inductive step: assume the statement holds for some natural number n, and prove that then the statement holds for n + 1.
The hypothesis in the inductive step, that the statement holds for some n, is called the induction hypothesis or inductive hypothesis. To prove the inductive step, one assumes the induction hypothesis and then uses this assumption, involving n, to prove the statement for n + 1.
Whether n = 0 or n = 1 is taken as the standard base case depends on the preferred definition of the natural numbers. In the fields of combinatorics and mathematical logic it is common to consider 0 as a natural number
Description
The simplest and most common form of mathematical induction infers that a statement involving a natural number n holds for all values of n. The proof consists of two steps:
The base case: prove that the statement holds for the first natural number n. Usually, n= 0 or n = 1; rarely, but sometimes conveniently, the base value of n may be taken as a larger number, or even as a negative number (the statement only holds at and above that threshold), because these extensions do not disturb the property of being a well-ordered set).The step case or inductive step: assume the statement holds for some natural number n, and prove that then the statement holds for n + 1.
The hypothesis in the inductive step, that the statement holds for some n, is called the induction hypothesis or inductive hypothesis. To prove the inductive step, one assumes the induction hypothesis and then uses this assumption, involving n, to prove the statement for n + 1.
Whether n = 0 or n = 1 is taken as the standard base case depends on the preferred definition of the natural numbers. In the fields of combinatorics and mathematical logic it is common to consider 0 as a natural number
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Mathematical induction can be used to prove that the following statement, P(n), holds for all natural numbers n. P(n) gives a formula for the sum of the natural numbers less than or equal to number n. ... Inductive step: Show that if P(k) holds, then also P(k + 1) holds.
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