induction

induction - NOTES FOR MA017 ON INDUCTION AND PIGEONHOLE...

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Unformatted text preview: NOTES FOR MA017 ON INDUCTION AND PIGEONHOLE PAUL NELSON 1. Induction 1.1. Basic definition and use. Induction , a line of reasoning used to prove statements about integers, goes as follows. Let a be an integer, and P ( n ) a statement about n for each n a . If (1) P ( a ) is true, and (2) for each k a , the truth of P ( k ) implies that of P ( k + 1), then P ( n ) is true for all integers n a . Example 1.1. Let n be a positive integer. Prove that 1 + 2 + + n = n ( n + 1) 2 . (1) P (1) is true: 1 = 1 (1+1) 2 . (2) Assume that P ( n ) is true, i.e. that 1 + 2 + + n = n ( n +1) 2 . Then 1 + 2 + + n + ( n + 1) = n ( n + 1) 2 + ( n + 1) = ( n + 1)( n + 2) 2 , i.e. P ( n + 1) is true. Example 1.2. Let n be a positive integer. Recall that n k = n ! k !( n- k )! , n = n n = 1 , n k- 1 + n k = n + 1 k . Prove that ( a + b ) n = n X k =0 n k a k b n- k . (1) ( a + b ) 1 = ( 1 ) a + ( 1 1 ) b . (2) Assume that ( a + b ) n = n k =0 ( n k ) a k b n- k . Multiplying both sides by ( a + b ), we obtain ( a + b ) k +1 = n X k =0 n k ( a + b ) a k b n- k = n +1 X k =1 n k- 1 a k b n +1- k + n X k =0 n k a k b n +1- k = n a n n +1 + n n a n +1 b + n X k =1 n k + n k- 1 a k b n +1- k = n +1 X k =0 n + 1 k a k b n +1- k . 1 2 PAUL NELSON 1.2. Working backwards with induction. It it often convenient to work backwards from P ( n + 1) to exploit the assumed truth of P ( n ). Example 1.3. Prove that f ( n ) = n 5 5 + n 4 2 + n 3 3- n 3 is an integer for n = 0 , 1 , 2 , 3 ,... . (1) f (0) = 0. (2) Assume f ( n ) is an integer. Then f ( n + 1) = n 5 + 5 n 4 + 10 n 3 + 10 n 2 + 5 n + 1 5 + n 4 + 4 k 3 + 6 k 2 + 4 k + 1 2 + n 3 + 3 n 2 + 3 n + 1 3- n + 1 30 ....
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induction - NOTES FOR MA017 ON INDUCTION AND PIGEONHOLE...

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