5.2 mathematical induction

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5.2 mathematical induction

  1. 1. Mathematical Induction<br />
  2. 2. Mathematical Induction<br />Mathematical induction is a method for verifying infinitely many (related) mathematical statements.<br />
  3. 3. Mathematical Induction<br />Mathematical induction is a method for verifying infinitely many (related) mathematical statements.<br />In this section, we use induction to verify formulas for sums and inequalities.<br />
  4. 4. Mathematical Induction<br />Mathematical induction is a method for verifying infinitely many (related) mathematical statements.<br />In this section, we use induction to verify formulas for sums and inequalities.<br />If we add consecutive odd numbers starting from 1: <br />
  5. 5. Mathematical Induction<br />Mathematical induction is a method for verifying infinitely many (related) mathematical statements.<br />In this section, we use induction to verify formulas for sums and inequalities.<br />If we add consecutive odd numbers starting from 1: <br />1 = 1<br />
  6. 6. Mathematical Induction<br />Mathematical induction is a method for verifying infinitely many (related) mathematical statements.<br />In this section, we use induction to verify formulas for sums and inequalities.<br />If we add consecutive odd numbers starting from 1: <br />1 = 1<br />1 + 3 = 4<br />
  7. 7. Mathematical Induction<br />Mathematical induction is a method for verifying infinitely many (related) mathematical statements.<br />In this section, we use induction to verify formulas for sums and inequalities.<br />If we add consecutive odd numbers starting from 1: <br />1 = 1<br />1 + 3 = 4<br />1 + 3 + 5 = 9 <br />
  8. 8. Mathematical Induction<br />Mathematical induction is a method for verifying infinitely many (related) mathematical statements.<br />In this section, we use induction to verify formulas for sums and inequalities.<br />If we add consecutive odd numbers starting from 1: <br />1 = 1<br />1 + 3 = 4<br />1 + 3 + 5 = 9 <br />1 + 3 + 5 + 7 = 16 <br />
  9. 9. Mathematical Induction<br />Mathematical induction is a method for verifying infinitely many (related) mathematical statements.<br />In this section, we use induction to verify formulas for sums and inequalities.<br />If we add consecutive odd numbers starting from 1: <br />1 = 1<br />1 + 3 = 4<br />1 + 3 + 5 = 9 <br />1 + 3 + 5 + 7 = 16 <br />We observe the following pattern: the sum of first n odd numbers is n2. <br />
  10. 10. Mathematical Induction<br />Mathematical induction is a method for verifying infinitely many (related) mathematical statements.<br />In this section, we use induction to verify formulas for sums and inequalities.<br />If we add consecutive odd numbers starting from 1: <br />1 = 1<br />1 + 3 = 4<br />1 + 3 + 5 = 9 <br />1 + 3 + 5 + 7 = 16 <br />We observe the following pattern: the sum of first n odd numbers is n2. Using the formula (2k – 1) for the k'th odd numbers, we write the above statement in <br />-notation as<br />
  11. 11. Mathematical Induction<br />Mathematical induction is a method for verifying infinitely many (related) mathematical statements.<br />In this section, we use induction to verify formulas for sums and inequalities.<br />If we add consecutive odd numbers starting from 1: <br />1 = 1<br />1 + 3 = 4<br />1 + 3 + 5 = 9 <br />1 + 3 + 5 + 7 = 16 <br />We observe the following pattern: the sum of first n odd numbers is n2. Using the formula (2k – 1) for the k'th odd numbers, we write the above statement in <br />-notation as "<br />n<br />(2k – 1) = n2 for all natural numbers n". <br />k=1<br />
  12. 12. Mathematical Induction<br />Two observations about the above formula.<br />
  13. 13. Mathematical Induction<br />Two observations about the above formula.<br />I. The formula actually consists of infinitely many statements (assertions), one for each number n, <br />
  14. 14. Mathematical Induction<br />Two observations about the above formula.<br />I. The formula actually consists of infinitely many statements (assertions), one for each number n, i.e. <br />we claim that all the following statements are true.<br />
  15. 15. Mathematical Induction<br />Two observations about the above formula.<br />I. The formula actually consists of infinitely many statements (assertions), one for each number n, i.e. <br />we claim that all the following statements are true.<br />S1: The sum of the first one odd number is 12 or that<br />1 = 12.<br />
  16. 16. Mathematical Induction<br />Two observations about the above formula.<br />I. The formula actually consists of infinitely many statements (assertions), one for each number n, i.e. <br />we claim that all the following statements are true.<br />S1: The sum of the first one odd number is 12 or that<br />1 = 12.<br />S2: The sum of the first two odd number is 22 or that<br />1 + 3 = 22.<br />
  17. 17. Mathematical Induction<br />Two observations about the above formula.<br />I. The formula actually consists of infinitely many statements (assertions), one for each number n, i.e. <br />we claim that all the following statements are true.<br />S1: The sum of the first one odd number is 12 or that<br />1 = 12.<br />S2: The sum of the first two odd number is 22 or that<br />1 + 3 = 22.<br />S3: The sum of the first three odd number is 32 or that<br />1 + 3 + 5 = 32.<br />
  18. 18. Mathematical Induction<br />Two observations about the above formula.<br />I. The formula actually consists of infinitely many statements (assertions), one for each number n, i.e. <br />we claim that all the following statements are true.<br />S1: The sum of the first one odd number is 12 or that<br />1 = 12.<br />S2: The sum of the first two odd number is 22 or that<br />1 + 3 = 22.<br />S3: The sum of the first three odd number is 32 or that<br />1 + 3 + 5 = 32.<br />.<br />.<br />Sn: The sum of the first n odd numbers is n2 or that<br />1 + 3 + 5 + … + (2n – 1) = n2.<br />.<br />.<br />.<br />
  19. 19. Mathematical Induction<br />II. We may verify the first few statements easily, but it is impossible to check all of them one by one. <br />
  20. 20. Mathematical Induction<br />II. We may verify the first few statements easily, but it is impossible to check all of them one by one. <br />Mathematical induction is a method of verifying all of them without actually checking every one of them. <br />
  21. 21. Mathematical Induction<br />II. We may verify the first few statements easily, but it is impossible to check all of them one by one. <br />Mathematical induction is a method of verifying all of them without actually checking every one of them. <br />Principle of Mathematics Induction:<br />
  22. 22. Mathematical Induction<br />II. We may verify the first few statements easily, but it is impossible to check all of them one by one. <br />Mathematical induction is a method of verifying all of them without actually checking every one of them. <br />Principle of Mathematics Induction:<br />Given infinitely many statements S1, S2, S3,.. <br />
  23. 23. Mathematical Induction<br />II. We may verify the first few statements easily, but it is impossible to check all of them one by one. <br />Mathematical induction is a method of verifying all of them without actually checking every one of them. <br />Principle of Mathematics Induction:<br />Given infinitely many statements S1, S2, S3,.. and that<br />1. S1 is true,<br />
  24. 24. Mathematical Induction<br />II. We may verify the first few statements easily, but it is impossible to check all of them one by one. <br />Mathematical induction is a method of verifying all of them without actually checking every one of them. <br />Principle of Mathematics Induction:<br />Given infinitely many statements S1, S2, S3,.. and that<br />1. S1 is true,<br />2. If SN is true implies that SN+1 must also be true,<br />
  25. 25. Mathematical Induction<br />II. We may verify the first few statements easily, but it is impossible to check all of them one by one. <br />Mathematical induction is a method of verifying all of them without actually checking every one of them. <br />Principle of Mathematics Induction:<br />Given infinitely many statements S1, S2, S3,.. and that<br />1. S1 is true,<br />2. If SN is true implies that SN+1 must also be true,<br />then all the statements are true. <br />
  26. 26. Mathematical Induction<br />II. We may verify the first few statements easily, but it is impossible to check all of them one by one. <br />Mathematical induction is a method of verifying all of them without actually checking every one of them. <br />Principle of Mathematics Induction:<br />Given infinitely many statements S1, S2, S3,.. and that<br />1. S1 is true,<br />2. If SN is true implies that SN+1 must also be true,<br />then all the statements are true. <br />This principle is analogous to the "dominos effect".<br />
  27. 27. Mathematical Induction<br />II. We may verify the first few statements easily, but it is impossible to check all of them one by one. <br />Mathematical induction is a method of verifying all of them without actually checking every one of them. <br />Principle of Mathematics Induction:<br />Given infinitely many statements S1, S2, S3,.. and that<br />1. S1 is true,<br />2. If SN is true implies that SN+1 must also be true,<br />then all the statements are true. <br />This principle is analogous to the "dominos effect". <br />If infinitely many dominos are lined up, two things <br />must be verified to make sure they all fall down.<br />
  28. 28. Mathematical Induction<br />II. We may verify the first few statements easily, but it is impossible to check all of them one by one. <br />Mathematical induction is a method of verifying all of them without actually checking every one of them. <br />Principle of Mathematics Induction:<br />Given infinitely many statements S1, S2, S3,.. and that<br />1. S1 is true,<br />2. If SN is true implies that SN+1 must also be true,<br />then all the statements are true. <br />This principle is analogous to the "dominos effect". <br />If infinitely many dominos are lined up, two things <br />must be verified to make sure they all fall down.<br />1. the first domino falls. <br />
  29. 29. Mathematical Induction<br />II. We may verify the first few statements easily, but it is impossible to check all of them one by one. <br />Mathematical induction is a method of verifying all of them without actually checking every one of them. <br />Principle of Mathematics Induction:<br />Given infinitely many statements S1, S2, S3,.. and that<br />1. S1 is true,<br />2. If SN is true implies that SN+1 must also be true,<br />then all the statements are true. <br />This principle is analogous to the "dominos effect". <br />If infinitely many dominos are lined up, two things <br />must be verified to make sure they all fall down.<br />1. the first domino falls. <br />2. the fall of the any domino will cause next one to fall. <br />
  30. 30. Mathematical Induction<br />To use induction to verify statements, we always<br />0. declare that induction method is used,<br />
  31. 31. Mathematical Induction<br />To use induction to verify statements, we always<br />0. declare that induction method is used,<br />1. verify the first statement S1 is true,<br />
  32. 32. Mathematical Induction<br />To use induction to verify statements, we always<br />0. declare that induction method is used,<br />1. verify the first statement S1 is true,<br />2. assume the N'th statement SN is true, <br />
  33. 33. Mathematical Induction<br />To use induction to verify statements, we always<br />0. declare that induction method is used,<br />1. verify the first statement S1 is true,<br />2. assume the N'th statement SN is true, use this to verify that as a consequence, SN+1 must also be true. <br />
  34. 34. Mathematical Induction<br />To use induction to verify statements, we always<br />0. declare that induction method is used,<br />1. verify the first statement S1 is true,<br />2. assume the N'th statement SN is true, use this to verify that as a consequence, SN+1 must also be true. <br />These steps are estabished format in mathemtics and they should be followed. <br />
  35. 35. Mathematical Induction<br />To use induction to verify statements, we always<br />0. declare that induction method is used,<br />1. verify the first statement S1 is true,<br />2. assume the N'th statement SN is true, use this to verify that as a consequence, SN+1 must also be true. <br />These steps are estabished format in mathemtics and they should be followed. <br />Example A: <br />Verify that (2k – 1) = n2 for all natural numbers n. <br />n<br />k=1<br />
  36. 36. Mathematical Induction<br />To use induction to verify statements, we always<br />0. declare that induction method is used,<br />1. verify the first statement S1 is true,<br />2. assume the N'th statement SN is true, use this to verify that as a consequence, SN+1 must also be true. <br />These steps are estabished format in mathemtics and they should be followed. <br />Example A: <br />Verify that (2k – 1) = n2 for all natural numbers n. <br />n<br />k=1<br />We will use induction. (declaring the methodology)<br />
  37. 37. Mathematical Induction<br />To use induction to verify statements, we always<br />0. declare that induction method is used,<br />1. verify the first statement S1 is true,<br />2. assume the N'th statement SN is true, use this to verify that as a consequence, SN+1 must also be true. <br />These steps are estabished format in mathemtics and they should be followed. <br />Example A: <br />Verify that (2k – 1) = n2 for all natural numbers n. <br />n<br />k=1<br />We will use induction. (declaring the methodology)<br />I. Verify its true when n = 1. (i.e S1 is true )<br />
  38. 38. Mathematical Induction<br />To use induction to verify statements, we always<br />0. declare that induction method is used,<br />1. verify the first statement S1 is true,<br />2. assume the N'th statement SN is true, use this to verify that as a consequence, SN+1 must also be true. <br />These steps are estabished format in mathemtics and they should be followed. <br />Example A: <br />Verify that (2k – 1) = n2 for all natural numbers n. <br />n<br />k=1<br />We will use induction. (declaring the methodology)<br />I. Verify its true when n = 1. (i.e S1 is true )<br />If n = 1, we've<br />
  39. 39. Mathematical Induction<br />To use induction to verify statements, we always<br />0. declare that induction method is used,<br />1. verify the first statement S1 is true,<br />2. assume the N'th statement SN is true, use this to verify that as a consequence, SN+1 must also be true. <br />These steps are estabished format in mathemtics and they should be followed. <br />Example A: <br />Verify that (2k – 1) = n2 for all natural numbers n. <br />n<br />k=1<br />We will use induction. (declaring the methodology)<br />I. Verify its true when n = 1. (i.e S1 is true )<br />1<br />(2k – 1) <br />If n = 1, we've<br />k=1<br />
  40. 40. Mathematical Induction<br />To use induction to verify statements, we always<br />0. declare that induction method is used,<br />1. verify the first statement S1 is true,<br />2. assume the N'th statement SN is true, use this to verify that as a consequence, SN+1 must also be true. <br />These steps are estabished format in mathemtics and they should be followed. <br />Example A: <br />Verify that (2k – 1) = n2 for all natural numbers n. <br />n<br />k=1<br />We will use induction. (declaring the methodology)<br />I. Verify its true when n = 1. (i.e S1 is true )<br />1<br />(2k – 1) = 1 <br />If n = 1, we've<br />k=1<br />
  41. 41. Mathematical Induction<br />To use induction to verify statements, we always<br />0. declare that induction method is used,<br />1. verify the first statement S1 is true,<br />2. assume the N'th statement SN is true, use this to verify that as a consequence, SN+1 must also be true. <br />These steps are estabished format in mathemtics and they should be followed. <br />Example A: <br />Verify that (2k – 1) = n2 for all natural numbers n. <br />n<br />k=1<br />We will use induction. (declaring the methodology)<br />I. Verify its true when n = 1. (i.e S1 is true )<br />1<br />(2k – 1) = 1 = 12. Done.<br />If n = 1, we've<br />k=1<br />
  42. 42. Mathematical Induction<br />II. Assume the formula works for n = N. <br />
  43. 43. Mathematical Induction<br />II. Assume the formula works for n = N. <br />N<br />(This means that (2k – 1) = 1 + 3 + .. + (2N – 1) = N2is true.)<br />k=1<br />
  44. 44. Mathematical Induction<br />II. Assume the formula works for n = N. <br />N<br />(This means that (2k – 1) = 1 + 3 + .. + (2N – 1) = N2is true.)<br />k=1<br />To verify it follows the formula will work for n = N+1.<br />
  45. 45. Mathematical Induction<br />II. Assume the formula works for n = N. <br />N<br />(This means that (2k – 1) = 1 + 3 + .. + (2N – 1) = N2is true.)<br />k=1<br />To verify it follows the formula will work for n = N+1.<br />N+1<br />(i.e. to show (2k – 1) =(N+1)2)<br />k=1<br />
  46. 46. Mathematical Induction<br />II. Assume the formula works for n = N. <br />N<br />(This means that (2k – 1) = 1 + 3 + .. + (2N – 1) = N2is true.)<br />k=1<br />To verify it follows the formula will work for n = N+1.<br />N+1<br />(i.e. to show (2k – 1) =(N+1)2)<br />k=1<br />For n = N+1, <br />
  47. 47. Mathematical Induction<br />II. Assume the formula works for n = N. <br />N<br />(This means that (2k – 1) = 1 + 3 + .. + (2N – 1) = N2is true.)<br />k=1<br />To verify it follows the formula will work for n = N+1.<br />N+1<br />(i.e. to show (2k – 1) =(N+1)2)<br />k=1<br />For n = N+1, the left hand sum is<br />N+1<br />(2k – 1) <br />k=1<br />
  48. 48. Mathematical Induction<br />II. Assume the formula works for n = N. <br />N<br />(This means that (2k – 1) = 1 + 3 + .. + (2N – 1) = N2is true.)<br />k=1<br />To verify it follows the formula will work for n = N+1.<br />N+1<br />(i.e. to show (2k – 1) =(N+1)2)<br />k=1<br />For n = N+1, the left hand sum is<br />N+1<br />(2k – 1) = 1 + 3 + .. + (2N – 1) + [2(N+1) – 1]<br />k=1<br />
  49. 49. Mathematical Induction<br />II. Assume the formula works for n = N. <br />N<br />(This means that (2k – 1) = 1 + 3 + .. + (2N – 1) = N2is true.)<br />k=1<br />To verify it follows the formula will work for n = N+1.<br />N+1<br />(i.e. to show (2k – 1) =(N+1)2)<br />k=1<br />For n = N+1, the left hand sum is<br />N+1<br />(2k – 1) = 1 + 3 + .. + (2N – 1) + [2(N+1) – 1]<br />k=1<br />by the induction assumation this sum is N2, <br />
  50. 50. Mathematical Induction<br />II. Assume the formula works for n = N. <br />N<br />(This means that (2k – 1) = 1 + 3 + .. + (2N – 1) = N2is true.)<br />k=1<br />To verify it follows the formula will work for n = N+1.<br />N+1<br />(i.e. to show (2k – 1) =(N+1)2)<br />k=1<br />For n = N+1, the left hand sum is<br />N+1<br />(2k – 1) = 1 + 3 + .. + (2N – 1) + [2(N+1) – 1]<br />k=1<br />by the induction assumation this sum is N2, hence<br /> = N2 + [2(N+1) – 1]<br />
  51. 51. Mathematical Induction<br />II. Assume the formula works for n = N. <br />N<br />(This means that (2k – 1) = 1 + 3 + .. + (2N – 1) = N2is true.)<br />k=1<br />To verify it follows the formula will work for n = N+1.<br />N+1<br />(i.e. to show (2k – 1) =(N+1)2)<br />k=1<br />For n = N+1, the left hand sum is<br />N+1<br />(2k – 1) = 1 + 3 + .. + (2N – 1) + [2(N+1) – 1]<br />k=1<br />by the induction assumation this sum is N2, hence<br /> = N2 + [2(N+1) – 1]<br /> = N2 + 2N + 1<br />
  52. 52. Mathematical Induction<br />II. Assume the formula works for n = N. <br />N<br />(This means that (2k – 1) = 1 + 3 + .. + (2N – 1) = N2is true.)<br />k=1<br />To verify it follows the formula will work for n = N+1.<br />N+1<br />(i.e. to show (2k – 1) =(N+1)2)<br />k=1<br />For n = N+1, the left hand sum is<br />N+1<br />(2k – 1) = 1 + 3 + .. + (2N – 1) + [2(N+1) – 1]<br />k=1<br />by the induction assumation this sum is N2, hence<br /> = N2 + [2(N+1) – 1]<br /> = N2 + 2N + 1<br /> = (N + 1)2 Done.<br />
  53. 53. Mathematical Induction<br />II. Assume the formula works for n = N. <br />N<br />(This means that (2k – 1) = 1 + 3 + .. + (2N – 1) = N2is true.)<br />k=1<br />To verify it follows the formula will work for n = N+1.<br />N+1<br />(i.e. to show (2k – 1) =(N+1)2)<br />k=1<br />For n = N+1, the left hand sum is<br />N+1<br />(2k – 1) = 1 + 3 + .. + (2N – 1) + [2(N+1) – 1]<br />k=1<br />by the induction assumation this sum is N2, hence<br /> = N2 + [2(N+1) – 1]<br /> = N2 + 2N + 1<br /> = (N + 1)2 Done.<br />n<br />Therefore (2k – 1) = n2 for all natural numbers n. <br />k=1<br />
  54. 54. Mathematical Induction<br />Example B: <br />Verify that n + 1 < 2n for all numbers n =1, 2, 3,…<br />
  55. 55. Mathematical Induction<br />Example B: <br />Verify that n + 1 < 2n for all numbers n =1, 2, 3,…<br />We use induction to verify this.<br />
  56. 56. Mathematical Induction<br />Example B: <br />Verify that n + 1 < 2n for all numbers n =1, 2, 3,…<br />We use induction to verify this.<br />I. Verify its true when n = 1. <br />
  57. 57. Mathematical Induction<br />Example B: <br />Verify that n + 1 < 2n for all numbers n =1, 2, 3,…<br />We use induction to verify this.<br />I. Verify its true when n = 1. <br />If n = 1, <br />
  58. 58. Mathematical Induction<br />Example B: <br />Verify that n + 1 < 2n for all numbers n =1, 2, 3,…<br />We use induction to verify this.<br />I. Verify its true when n = 1. <br />If n = 1, we check that its true that 1 + 1 < 21.<br />
  59. 59. Mathematical Induction<br />Example B: <br />Verify that n + 1 < 2n for all numbers n =1, 2, 3,…<br />We use induction to verify this.<br />I. Verify its true when n = 1. <br />If n = 1, we check that its true that 1 + 1 < 21.<br />II. Assume the ineqality is true for n = N, <br />
  60. 60. Mathematical Induction<br />Example B: <br />Verify that n + 1 < 2n for all numbers n =1, 2, 3,…<br />We use induction to verify this.<br />I. Verify its true when n = 1. <br />If n = 1, we check that its true that 1 + 1 < 21.<br />II. Assume the ineqality is true for n = N, that is, its true that N + 1 < 2N , <br />
  61. 61. Mathematical Induction<br />Example B: <br />Verify that n + 1 < 2n for all numbers n =1, 2, 3,…<br />We use induction to verify this.<br />I. Verify its true when n = 1. <br />If n = 1, we check that its true that 1 + 1 < 21.<br />II. Assume the ineqality is true for n = N, that is, its true that N + 1 < 2N , to verify that its has to be true when n = N + 1, <br />
  62. 62. Mathematical Induction<br />Example B: <br />Verify that n + 1 < 2n for all numbers n =1, 2, 3,…<br />We use induction to verify this.<br />I. Verify its true when n = 1. <br />If n = 1, we check that its true that 1 + 1 < 21.<br />II. Assume the ineqality is true for n = N, that is, its true that N + 1 < 2N , to verify that its has to be true when n = N + 1, i.e (N + 1) + 1 < 2N+1.<br />
  63. 63. Mathematical Induction<br />Example B: <br />Verify that n + 1 < 2n for all numbers n =1, 2, 3,…<br />We use induction to verify this.<br />I. Verify its true when n = 1. <br />If n = 1, we check that its true that 1 + 1 < 21.<br />II. Assume the ineqality is true for n = N, that is, its true that N + 1 < 2N , to verify that its has to be true when n = N + 1, i.e (N + 1) + 1 < 2N+1.<br />When n = N + 1, the left hand side is<br />(N + 1) + 1 <br />
  64. 64. Mathematical Induction<br />Example B: <br />Verify that n + 1 < 2n for all numbers n =1, 2, 3,…<br />We use induction to verify this.<br />I. Verify its true when n = 1. <br />If n = 1, we check that its true that 1 + 1 < 21.<br />II. Assume the ineqality is true for n = N, that is, its true that N + 1 < 2N , to verify that its has to be true when n = N + 1, i.e (N + 1) + 1 < 2N+1.<br />When n = N + 1, the left hand side is<br />(N + 1) + 1 <br />by the induction assumation this is < 2N<br />
  65. 65. Mathematical Induction<br />Example B: <br />Verify that n + 1 < 2n for all numbers n =1, 2, 3,…<br />We use induction to verify this.<br />I. Verify its true when n = 1. <br />If n = 1, we check that its true that 1 + 1 < 21.<br />II. Assume the ineqality is true for n = N, that is, its true that N + 1 < 2N , to verify that its has to be true when n = N + 1, i.e (N + 1) + 1 < 2N+1.<br />When n = N + 1, the left hand side is<br />(N + 1) + 1 < 2N + 1<br />by the induction assumation this is < 2N<br />
  66. 66. Mathematical Induction<br />Example B: <br />Verify that n + 1 < 2n for all numbers n =1, 2, 3,…<br />We use induction to verify this.<br />I. Verify its true when n = 1. <br />If n = 1, we check that its true that 1 + 1 < 21.<br />II. Assume the ineqality is true for n = N, that is, its true that N + 1 < 2N , to verify that its has to be true when n = N + 1, i.e (N + 1) + 1 < 2N+1.<br />When n = N + 1, the left hand side is<br />(N + 1) + 1 < 2N + 1< 2N + 2N<br />by the induction assumation this is < 2N<br />
  67. 67. Mathematical Induction<br />Example B: <br />Verify that n + 1 < 2n for all numbers n =1, 2, 3,…<br />We use induction to verify this.<br />I. Verify its true when n = 1. <br />If n = 1, we check that its true that 1 + 1 < 21.<br />II. Assume the ineqality is true for n = N, that is, its true that N + 1 < 2N , to verify that its has to be true when n = N + 1, i.e (N + 1) + 1 < 2N+1.<br />When n = N + 1, the left hand side is<br />(N + 1) + 1 < 2N + 1< 2N + 2N < 2*2N<br />by the induction assumation this is < 2N<br />
  68. 68. Mathematical Induction<br />Example B: <br />Verify that n + 1 < 2n for all numbers n =1, 2, 3,…<br />We use induction to verify this.<br />I. Verify its true when n = 1. <br />If n = 1, we check that its true that 1 + 1 < 21.<br />II. Assume the ineqality is true for n = N, that is, its true that N + 1 < 2N , to verify that its has to be true when n = N + 1, i.e (N + 1) + 1 < 2N+1.<br />When n = N + 1, the left hand side is<br />(N + 1) + 1 < 2N + 1< 2N + 2N < 2*2N < 2N+1. Done.<br />by the induction assumation this is < 2N<br />
  69. 69. Properties Summation Notation<br />Although the induction method gives us a way to verify formulas, it does not gives us any insight where the formulas came from. <br />
  70. 70. Properties Summation Notation<br />Although the induction method gives us a way to verify formulas, it does not gives us any insight where the formulas came from. <br />In mathematics, its always the case that after we observe certain pattern first, then the induction method is used to verify our observation.<br />

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