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# induction - Intermediate Mathematics Proof by Induction R...

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Intermediate Mathematics Proof by Induction R Horan & M Lavelle The aim of this package is to provide a short self assessment programme for students who want to understand the method of proof by induction. Copyright c 2005 Last Revision Date: February 12, 2006 Version 1.0

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Table of Contents 1. Introduction (Summation) 2. The Principle of Induction 3. Further Examples 4. Final Quiz Solutions to Exercises Solutions to Quizzes The full range of these packages and some instructions, should they be required, can be obtained from our web page Mathematics Support Materials.
Section 1: Introduction (Summation) 3 1. Introduction (Summation) Proof by induction involves statements which depend on the natural numbers, n = 1 , 2 , 3 , . . . . It often uses summation notation which we now briefly review before discussing induction itself. We write the sum of the natural numbers up to a value n as: 1 + 2 + 3 + · · · + ( n - 1) + n = n i =1 i . The symbol denotes a sum over its argument for each natural number i from the lowest value, here i = 1 , to the maximum value, here i = n . Example 1: Write out explicitly the following sums: a) 6 i =3 i , b) 3 i =1 (2 i + 1) , c) 4 i =1 2 i .

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Section 1: Introduction (Summation) 4 The above sums when written out are: a) 6 i =3 i = 3 + 4 + 5 + 6 , b) 3 i =1 (2 i + 1) = (2 × 1 + 1) + (2 × 2 + 1) + (2 × 3 + 1) = 3 + 5 + 7 , c) 4 i =1 2 i = 2 1 + 2 2 + 2 3 + 2 4 . It is important to realise that the choice of symbol for the variable we are summing over is arbitrary, e.g., the following two sums are identical: 4 i =1 i 3 = 4 j =1 j 3 = 1 3 + 2 3 + 3 3 + 4 3 . The variable that is summed over is called a dummy variable .
Section 1: Introduction (Summation) 5 Quiz Select from the answers below the value of 5 i =2 2 i . (a) 1024 , (b) 62 , (c) 60 , (d) 32 . Exercise 1. Expand the sums (click on the green letters for solutions): (a) 3 i =1 (2 i - 1) , (b) 4 j =1 (2 j - 1) , (c) 4 s =1 10 s , (d) 3 j =1 12 j , (e) 3 i =0 3(2 i + 1) , (f) 3 j =1 1 j 2 . Exercise 2. Express the following in summation notation. (a) 1 + 20 + 400 + 8 , 000 , (b) - 3 - 1 + 1 + 3 + 5 + 7 . Hint: write a) as a sum of powers of 20 .

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Section 2: The Principle of Induction 6 2. The Principle of Induction Induction is an extremely powerful method of proving results in many areas of mathematics. It is based upon the following principle. The Induction Principle: let P ( n ) be a statement which involves a natural number n , i.e., n = 1 , 2 , 3 . . . , then P ( n ) is true for all n if a) P (1) is true, and b) P ( k ) P ( k + 1) for all natural numbers k . The standard analogy to this involves a row of dominoes: if it is shown that toppling one domino will make the next fall over (step b) and that the first domino has fallen (step a) then it follows that all of the dominoes in the row will fall over. Example 2: the result of adding the first n natural numbers is: 1 + 2 + 3 + · · · + n = n ( n + 1) 2 ; i.e., P ( n ) is n i =1 i = n ( n + 1) 2 . This is proven on the next page.
Section 2: The Principle of Induction 7 Step a) (the check): for n = 1 , 1 i =1 i = 1 = 1 × 2 2 .

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induction - Intermediate Mathematics Proof by Induction R...

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