Mathematical Induction - Mathematical Induction Advanced...

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Mathematical Induction Advanced Level Pure Mathematics Advanced Level Pure Mathematics Algebra Chapter 3 Mathematical Induction 3.1 First Principle of Mathematical Induction 2 3.2 Second Principle of Mathematical Induction 9 3.1 First Principle of Mathematical Induction Prepared by K. F. Ngai Page 1 3
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Mathematical Induction Advanced Level Pure Mathematics Definition Let ) ( n P be a proposition on an integer variable n . Then ) ( n P is true for all integers s n if and only if the following two conditions are both satisfied : (i) ) ( s P is true , ii) If ) ( k P , where k s , is assumed to be true, then ) 1 ( + k P is true . Example 1 Prove that for all positive integers n , 3 5 5 5 5 3 3 3 3 ) 3 2 1 ( 4 ) 3 2 1 ( 3 ) 3 2 1 ( n n n + + + = + + + + + + + + + Example 2 Prove, by induction, that n n n n n 2 1 1 2 1 ... 4 1 3 1 2 1 1 2 1 ... 2 1 1 1 - - + + - + - = + + + + + Prepared by K. F. Ngai Page 2
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Advanced Level Pure Mathematics for all positive integers n . (1) When n = 1, L.H.S. = 2 1 R.H.S. = 2 1 2 1 1 = - Hence, the proposition is true for n = 1. (2) Assume the proposition is true for n = k , k 1. i.e. k k k k k 2 1 1 2 1 ... 4 1 3 1 2 1 1 2 1 ... 2 1 1 1 - - + + - + - = + + + + + When n = k + 1, L.H.S. = 1 1 1 ... 2 1 1 1 + + + + + + + k k = Hence, the proposition is also true for n = k + 1. By the Principle of Mathematical Induction, the proposition is true for all positive integers n . Example 3 A sequence a 1 , a 2 , , a n is defined as follows : a 1 = 1 , a 2 = 2 and a n +2 = a n +1 + a n for n 1. Prove, by induction, that
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Mathematical Induction - Mathematical Induction Advanced...

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