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X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
X2 T08 02 induction
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X2 T08 02 induction

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  • 1. Mathematical Induction
  • 2. Mathematical Induction 1 1 1 1 e.g. i  Prove 1  2  2    2  2  2 3 n n
  • 3. Mathematical Induction 1 1 1 1 e.g. i  Prove 1  2  2    2  2  2 3 n n Test: n = 1
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. (ii) A sequence is defined by; a1  2 an1  2  an for n  1 Show that an  2 for n  1
  • 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. (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. (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. (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. (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. (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. (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. (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. (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. (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. (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. (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. (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. (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. (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. (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. (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. (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. (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. Proof: ak 1  ak  ak 1
  • 47. Proof: ak 1  ak  ak 1 k k 1 1  5  1  5       2   2 
  • 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. 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. 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. 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. 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. 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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