X2 T08 02 induction

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X2 T08 02 induction

  1. 1. Mathematical Induction
  2. 2. Mathematical Induction 1 1 1 1 e.g. i  Prove 1  2  2    2  2  2 3 n n
  3. 3. Mathematical Induction 1 1 1 1 e.g. i  Prove 1  2  2    2  2  2 3 n n Test: n = 1
  4. 4. Mathematical Induction 1 1 1 1 e.g. i  Prove 1  2  2    2  2  2 3 n n 1 Test: n = 1 L.H .S  2 1 1
  5. 5. Mathematical Induction 1 1 1 1 e.g. i  Prove 1  2  2    2  2  2 3 n n 1 1 Test: n = 1 L.H .S  2 R.H .S  2  1 1 1 1
  6. 6. Mathematical Induction 1 1 1 1 e.g. i  Prove 1  2  2    2  2  2 3 n n 1 1 Test: n = 1 L.H .S  2 R.H .S  2  1 1 1 1  L.H .S  R.H .S
  7. 7. Mathematical Induction 1 1 1 1 e.g. i  Prove 1  2  2    2  2  2 3 n n 1 1 Test: n = 1 L.H .S  2 R.H .S  2  1 1 1 1  L.H .S  R.H .S 1 1 1 1 A n  k  1   2  2  2  22 3 k k
  8. 8. Mathematical Induction 1 1 1 1 e.g. i  Prove 1  2  2    2  2  2 3 n n 1 1 Test: n = 1 L.H .S  2 R.H .S  2  1 1 1 1  L.H .S  R.H .S 1 1 1 1 A n  k  1   2  2  2  22 3 k k 1 1 1 1 P n  k  1 1  2  2     2 2 3 k  12 k 1
  9. 9. Proof: 1 1 1 1 1 1 1 1  2   1 2  2  2  22 3 k  12 2 3 k k  12
  10. 10. Proof: 1 1 1 1 1 1 1 1  2   1 2  2  2  22 3 k  12 2 3 k k  12 1 1  2  k k  12
  11. 11. Proof: 1 1 1 1 1 1 1 1  2   1 2  2  2  22 3 k  12 2 3 k k  12 1 1  2  k k  12 k  1  k 2  2 k k  1 2
  12. 12. Proof: 1 1 1 1 1 1 1 1  2   1 2  2  2  22 3 k  12 2 3 k k  12 1 1  2  k k  12 k  1  k 2  2 k k  1 2 k 2  k 1  2 k k  1 2
  13. 13. Proof: 1 1 1 1 1 1 1 1  2   1 2  2  2  22 3 k  12 2 3 k k  12 1 1  2  k k  12 k  1  k 2  2 k k  1 2 k 2  k 1  2 k k  1 2 k2  k 1  2  k k  1 k k  1 2 2
  14. 14. Proof: 1 1 1 1 1 1 1 1  2   1 2  2  2  22 3 k  12 2 3 k k  12 1 1  2  k k  12 k  1  k 2  2 k k  1 2 k 2  k 1  2 k k  1 2 k2  k 1  2  k k  1 k k  1 2 2 k k  1  2 k k  1 2
  15. 15. Proof: 1 1 1 1 1 1 1 1  2   1 2  2  2  22 3 k  12 2 3 k k  12 1 1  2  k k  12 k  1  k 2  2 k k  1 2 k 2  k 1  2 k k  1 2 k2  k 1  2  k k  1 k k  1 2 2 k k  1  2 k k  1 2 1  2 k 1
  16. 16. Proof: 1 1 1 1 1 1 1 1  2   1 2  2  2  22 3 k  12 2 3 k k  12 1 1  2  k k  12 k  1  k 2  2 k k  1 2 k 2  k 1  2 k k  1 2 k2  k 1  2  k k  1 k k  1 2 2 k k  1  2 k k  1 2 1  2 k 1 1 1 1 1 1  2  2     2 2 3 k  12 k 1
  17. 17. (ii) A sequence is defined by; a1  2 an1  2  an for n  1 Show that an  2 for n  1
  18. 18. (ii) A sequence is defined by; a1  2 an1  2  an for n  1 Show that an  2 for n  1 Test: n = 1
  19. 19. (ii) A sequence is defined by; a1  2 an1  2  an for n  1 Show that an  2 for n  1 Test: n = 1 a1  2  2
  20. 20. (ii) A sequence is defined by; a1  2 an1  2  an for n  1 Show that an  2 for n  1 Test: n = 1 a1  2  2 A n  k  a k  2
  21. 21. (ii) A sequence is defined by; a1  2 an1  2  an for n  1 Show that an  2 for n  1 Test: n = 1 a1  2  2 A n  k  a k  2 P n  k  1 ak 1  2
  22. 22. (ii) A sequence is defined by; a1  2 an1  2  an for n  1 Show that an  2 for n  1 Test: n = 1 a1  2  2 A n  k  a k  2 P n  k  1 ak 1  2 Proof:
  23. 23. (ii) A sequence is defined by; a1  2 an1  2  an for n  1 Show that an  2 for n  1 Test: n = 1 a1  2  2 A n  k  a k  2 P n  k  1 ak 1  2 Proof: ak 1  2  ak
  24. 24. (ii) A sequence is defined by; a1  2 an1  2  an for n  1 Show that an  2 for n  1 Test: n = 1 a1  2  2 A n  k  a k  2 P n  k  1 ak 1  2 Proof: ak 1  2  ak  22
  25. 25. (ii) A sequence is defined by; a1  2 an1  2  an for n  1 Show that an  2 for n  1 Test: n = 1 a1  2  2 A n  k  a k  2 P n  k  1 ak 1  2 Proof: ak 1  2  ak  22  4 2
  26. 26. (ii) A sequence is defined by; a1  2 an1  2  an for n  1 Show that an  2 for n  1 Test: n = 1 a1  2  2 A n  k  a k  2 P n  k  1 ak 1  2 Proof: ak 1  2  ak  22  4 2  ak 1  2
  27. 27. iii  The sequences xn and yn are defined by; xn  y n 2 xn y n x1  5, y1  2 xn1  , yn1  2 xn  y n Prove xn yn  10 for n  1
  28. 28. iii  The sequences xn and yn are defined by; xn  y n 2 xn y n x1  5, y1  2 xn1  , yn1  2 xn  y n Prove xn yn  10 for n  1 Test: n = 1
  29. 29. iii  The sequences xn and yn are defined by; xn  y n 2 xn y n x1  5, y1  2 xn1  , yn1  2 xn  y n Prove xn yn  10 for n  1 Test: n = 1 x1 y1  52   10
  30. 30. iii  The sequences xn and yn are defined by; xn  y n 2 xn y n x1  5, y1  2 xn1  , yn1  2 xn  y n Prove xn yn  10 for n  1 Test: n = 1 x1 y1  52   10 A n  k  xk yk  10
  31. 31. iii  The sequences xn and yn are defined by; xn  y n 2 xn y n x1  5, y1  2 xn1  , yn1  2 xn  y n Prove xn yn  10 for n  1 Test: n = 1 x1 y1  52   10 A n  k  xk yk  10 P n  k  1 xk 1 yk 1  10
  32. 32. iii  The sequences xn and yn are defined by; xn  y n 2 xn y n x1  5, y1  2 xn1  , yn1  2 xn  y n Prove xn yn  10 for n  1 Test: n = 1 x1 y1  52   10 A n  k  xk yk  10 P n  k  1 xk 1 yk 1  10 Proof:
  33. 33. iii  The sequences xn and yn are defined by; xn  y n 2 xn y n x1  5, y1  2 xn1  , yn1  2 xn  y n Prove xn yn  10 for n  1 Test: n = 1 x1 y1  52   10 A n  k  xk yk  10 P n  k  1 xk 1 yk 1  10 Proof:  xk  yk  2 xk yk  xk 1 yk 1   x  y     2  k k 
  34. 34. iii  The sequences xn and yn are defined by; xn  y n 2 xn y n x1  5, y1  2 xn1  , yn1  2 xn  y n Prove xn yn  10 for n  1 Test: n = 1 x1 y1  52   10 A n  k  xk yk  10 P n  k  1 xk 1 yk 1  10 Proof:  xk  yk  2 xk yk  xk 1 yk 1   x  y     2  k k   xk y k  10
  35. 35. iii  The sequences xn and yn are defined by; xn  y n 2 xn y n x1  5, y1  2 xn1  , yn1  2 xn  y n Prove xn yn  10 for n  1 Test: n = 1 x1 y1  52   10 A n  k  xk yk  10 P n  k  1 xk 1 yk 1  10 Proof:  xk  yk  2 xk yk  xk 1 yk 1   x  y     2  k k   xk y k  10  xk 1 yk 1  10
  36. 36. (iv) The Fibonacci sequence is defined by; a1  a2  1 an1  an  an1 for n  1 n 1  5  Prove that an    for n  1  2 
  37. 37. (iv) The Fibonacci sequence is defined by; a1  a2  1 an1  an  an1 for n  1 n 1  5  Prove that an    for n  1  2  Test: n = 1 and n =2
  38. 38. (iv) The Fibonacci sequence is defined by; a1  a2  1 an1  an  an1 for n  1 n 1  5  Prove that an    for n  1  2  Test: n = 1 and n =2 L.H .S  a1 1
  39. 39. (iv) The Fibonacci sequence is defined by; a1  a2  1 an1  an  an1 for n  1 n 1  5  Prove that an    for n  1  2  Test: n = 1 and n =2 1 1  5  L.H .S  a1 R.H .S     2  1  1.62
  40. 40. (iv) The Fibonacci sequence is defined by; a1  a2  1 an1  an  an1 for n  1 n 1  5  Prove that an    for n  1  2  Test: n = 1 and n =2 1 1  5  L.H .S  a1 R.H .S     2  1  1.62  L.H .S  R.H .S
  41. 41. (iv) The Fibonacci sequence is defined by; a1  a2  1 an1  an  an1 for n  1 n 1  5  Prove that an    for n  1  2  Test: n = 1 and n =2 1 1  5  L.H .S  a1 R.H .S     2  1  1.62  L.H .S  R.H .S L.H .S  a2 1
  42. 42. (iv) The Fibonacci sequence is defined by; a1  a2  1 an1  an  an1 for n  1 n 1  5  Prove that an    for n  1  2  Test: n = 1 and n =2 1 1  5  L.H .S  a1 R.H .S     2  1  1.62  L.H .S  R.H .S 2 1  5  L.H .S  a2 R.H .S     2  1  2.62
  43. 43. (iv) The Fibonacci sequence is defined by; a1  a2  1 an1  an  an1 for n  1 n 1  5  Prove that an    for n  1  2  Test: n = 1 and n =2 1 1  5  L.H .S  a1 R.H .S     2  1  1.62  L.H .S  R.H .S 2 1  5  L.H .S  a2 R.H .S     2  1  2.62  L.H .S  R.H .S
  44. 44. (iv) The Fibonacci sequence is defined by; a1  a2  1 an1  an  an1 for n  1 n 1  5  Prove that an    for n  1  2  Test: n = 1 and n =2 1 1  5  L.H .S  a1 R.H .S     2  1  1.62  L.H .S  R.H .S 2 1  5  L.H .S  a2 R.H .S     2  1  2.62  L.H .S  R.H .S k 1 k 1  5  1  5  A n  k  1 & n  k  ak 1    & ak     2   2 
  45. 45. (iv) The Fibonacci sequence is defined by; a1  a2  1 an1  an  an1 for n  1 n 1  5  Prove that an    for n  1  2  Test: n = 1 and n =2 1 1  5  L.H .S  a1 R.H .S     2  1  1.62  L.H .S  R.H .S 2 1  5  L.H .S  a2 R.H .S     2  1  2.62  L.H .S  R.H .S k 1 k 1  5  1  5  A n  k  1 & n  k  ak 1    & ak     2   2  k 1 1 5  P n  k  1 ak 1      2 
  46. 46. Proof: ak 1  ak  ak 1
  47. 47. Proof: ak 1  ak  ak 1 k k 1 1  5  1  5       2   2 
  48. 48. Proof: ak 1  ak  ak 1 k k 1 1  5  1  5      2   2  k 1 1 2  1  5   1  5 1  5           2   2    2   
  49. 49. Proof: ak 1  ak  ak 1 k k 1 1  5  1 5      2   2  k 1 1 2  1  5   1  5 1  5           2   2    2    k 1 1  5   2 4      2  2  1  5 1  5  
  50. 50. Proof: ak 1  ak  ak 1 k k 1 1  5  1  5      2   2  k 1 1 2  1  5   1  5 1  5           2   2    2    k 1 1  5  2 4      2  2  1  5 1  5   k 1 1  5  2  2 5  4    2   2   1  5   k 1 1  5  62 5     2  2   1  5  
  51. 51. Proof: ak 1  ak  ak 1 k k 1 1  5  1  5      2   2  k 1 1 2  1  5   1  5 1  5           2   2    2    k 1 1  5  2 4      2  2  1  5 1  5   k 1 1  5  2  2 5  4    2   2   1  5   k 1 1  5   6  2 5       2   1  5 2  k 1 1  5     2 
  52. 52. Proof: ak 1  ak  ak 1 k k 1 1  5  1  5      2   2  k 1 1 2  1  5   1  5 1  5           2   2    2    k 1 1  5  2 4      2  2  1  5 1  5   k 1 1  5  2  2 5  4    2   2   1  5   k 1 1  5   6  2 5       2   1  5 2  k 1 1  5     2  k 1 1  5   ak 1     2 
  53. 53. Proof: ak 1  ak  ak 1 k k 1 1  5  1  5      2   2  k 1 1 2  1  5   1  5 1  5           2   2    2    k 1 1  5  2 4  Sheets     2  2  1  5 1  5   k 1 + 1  5  2  2 5  4    2   2   1  5   Exercise 10E* k 1 1  5   6  2 5       2   1  5 2  k 1 1  5     2  k 1 1  5   ak 1     2 

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