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mac1105_lecture26_2 - L26 Mathematical Induction Binomial...

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285 L26 Mathematical Induction; Binomial Theorem Mathematical Induction Mathematical induction is a method for proving mathematical statements which involve natural numbers. We denote a statement ( ) A n , n is natural. Example : ( ) A n : 2 4 6 ... (2 ) ( 1) n n n + + + + = + , 1, 2, 3, n = . The Principle of Mathematical Induction: Suppose that the following two conditions are satisfied with regard to a statement ( ) A n : 1. ( ) A n is true for 0 n n = . 2. If ( ) A n is true for n k = ( 0 k n ), it is also true for 1 n k = + . Then ( ) A n is true for all natural numbers 0 n n . 286 Example : Use mathematical induction to prove that the formulas are valid. ( ) A n : ( 1) 1 2 3 ... 2 n n n + + + + + = , 1 n
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