mac1105_lecture26_1

mac1105_lecture26_1 - 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) nn 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 = . 2. If ( ) A n is true for nk = ( 0 kn ), it is also true for 1 = + . Then ( ) A n is true for all natural numbers 0 .
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286 Example : Use mathematical induction to prove that the formulas are valid. () A n : (1 ) 123. . . 2 nn n + +++ += , 1 n
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287 () A n : 333 3 2 1 2 3 ... (1 2 3 . .. ) nn +++ += + + + + , 1 n
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288 We define !1 2 . . . nn = ⋅⋅ (1 ) ! n += Example : Use mathematical induction to prove that the statement is true for all natural numbers 3 n . () A n : 1 !2 n n >
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289 The Symbol n j ⎛⎞ ⎜⎟ ⎝⎠ If j and n are integers with 0 jn , then () ! !! n n jj n j = . Read: “ n taken j at a time”. 0
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mac1105_lecture26_1 - L26 Mathematical Induction; Binomial...

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