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Chapter 11: Further Topics in Algebra

Chapter 11: Further Topics in Algebra. 11.1 Sequences and Series 11.2 Arithmetic Sequences and Series 11.3 Geometric Sequences and Series 11.4 The Binomial Theorem 11.5 Mathematical Induction 11.6 Counting Theory 11.7 Probability. 11.5 Mathematical Induction.

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Chapter 11: Further Topics in Algebra

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  1. Chapter 11: Further Topics in Algebra 11.1 Sequences and Series 11.2 Arithmetic Sequences and Series 11.3 Geometric Sequences and Series 11.4 The Binomial Theorem 11.5 Mathematical Induction 11.6 Counting Theory 11.7 Probability

  2. 11.5 Mathematical Induction • Mathematical induction is used to prove statements claimed true for every positive integer n. For example, the summation rule is true for each integer n> 1.

  3. 11.5 Mathematical Induction Label the statement Sn. For any one value of n, the statement can be verified to be true.

  4. 11.5 Mathematical Induction To show Sn is true for every n requires mathematical induction.

  5. 11.5 Mathematical Induction Principle of Mathematical Induction Let Sn be a statement concerning the positive integer n. Suppose that 1. S1 is true; 2. For any positive integer k, k<n, if Sk is true, then Sk+1 is also true. Then, Sn is true for every positive integer n.

  6. 11.5 Mathematical Induction Proof by Mathematical Induction Step 1Prove that the statement is true for n = 1. Step 2 Show that for any positive integer k, if Sk is true, then Sk+1 is also true.

  7. 11.5 Proving an Equality Statement Example Let Sn be the statement Prove that Sn is true for every positive integer n. Solution The proof uses mathematical induction.

  8. 11.5 Proving an Equality Statement SolutionStep 1 Show that the statement is true when n = 1. S1 is the statement which is true since both sides equal 1.

  9. 11.5 Proving an Equality Statement SolutionStep 2 Show that if Sk is true then Sk+1 is also true. Start with Sk and assume it is a true statement. Add k + 1 to each side

  10. 11.5 Proving an Equality Statement SolutionStep 2

  11. 11.5 Proving an Equality Statement SolutionStep 2 This is the statement for n = k + 1. It has been shown that if Sk is true then Sk+1 is also true. By mathematical induction Sn is true for all positive integers n.

  12. 11.5 Mathematical Induction Generalized Principle of Mathematical Induction Let Sn be a statement concerning the positive integer n. Suppose that Step 1Sj is true; Step 2 For any positive integer k, k>j, if Sk implies Sk+1. Then, Sn is true for all positive integers n>j.

  13. 11.5 Using the Generalized Principle Example Let Sn represent the statement Show that Sn is true for all values of n> 3. Solution Since the statement is claimed to be true for values of n beginning with 3 and not 1, the proof uses the generalized principle of mathematical induction.

  14. 11.5 Using the Generalized Principle SolutionStep 1 Show that Sn is true when n = 3. S3 is the statement which is true since 8 > 7.

  15. 11.5 Using the Generalized Principle SolutionStep 2 Show that Sk implies Sk+1 for k > 3. Assume Sk is true. Multiply each side by 2, giving or or, equivalently

  16. 11.5 Using the Generalized Principle SolutionStep 2 Since k> 3, then 2k >1 and it follows that or which is the statement Sk+1. Thus Sk implies Sk+1 and, by the generalized principle, Sn is true for all n> 3.

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