week01 - MATHEMATICAL INDUCTION Original Notes adopted from...

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MATHEMATICAL INDUCTION Original Notes adopted from September 11, 2001(Week 1) c ± P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics The Natural Numbers The set of natural numbers is the set { 1 , 2 , 3 , 4 , 5 ,... } , which is denoted by N (or sometimes N .) Principle of Mathematical Induction If S N such that a) 1 S b) ( k + 1) S whenever k S , then S = N . Example. Prove: 1 2 + 2 2 + 3 2 + ... + n 2 = n ( n +1)(2 n +1) 6 . Let S = { m : 1 2 + 2 2 + 3 2 + ... + m 2 = m ( m +1)(2 m +1) 6 } . We need to show that S = N . By Mathematical Induction, it will suffice to show that a) 1 S b) ( k + 1) S whenever k S Is 1 S ? 1 2 = 1(1+1)(2+1) 6 = 1 Yes. Now suppose k S , that is, 1 2 + 2 2 + 3 2 + ... + k 2 = k ( k + 1)(2 k + 1) 6 Then we must show that 1 2 + 2 2 + 3 2 + ... + k 2 + ( k + 1) 2 = ( k + 1)( k + 2)(2 k + 2 + 1) 6 First, 1 2 + 2 2 + 3 2 + ... + k 2 = k ( k + 1)(2 k + 1) 6 , and so 1 2 + 2 2 + 3 2 + ... + k 2 + ( k + 1) 2 = k ( k + 1)(2 k + 1) 6 + ( k + 1) 2 = k ( k + 1)(2
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week01 - MATHEMATICAL INDUCTION Original Notes adopted from...

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