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# 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 