11X1 T15 01 applications of ap & gp

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  • 1. Applications of Arithmetic & Geometric Series
  • 2. Applications of Arithmetic & Geometric Series Formulae for Arithmetic Series
  • 3. Applications of Arithmetic & Geometric Series Formulae for Arithmetic Series A series is called an arithmetic series if; d  Tn  Tn1 where d is a constant, called the common difference
  • 4. Applications of Arithmetic & Geometric Series Formulae for Arithmetic Series A series is called an arithmetic series if; d  Tn  Tn1 where d is a constant, called the common difference The general term of an arithmetic series with first term a and common difference d is;
  • 5. Applications of Arithmetic & Geometric Series Formulae for Arithmetic Series A series is called an arithmetic series if; d  Tn  Tn1 where d is a constant, called the common difference The general term of an arithmetic series with first term a and common difference d is; Tn  a  n  1d
  • 6. Applications of Arithmetic & Geometric Series Formulae for Arithmetic Series A series is called an arithmetic series if; d  Tn  Tn1 where d is a constant, called the common difference The general term of an arithmetic series with first term a and common difference d is; Tn  a  n  1d Three numbers a, x and b are terms in an arithmetic series if;
  • 7. Applications of Arithmetic & Geometric Series Formulae for Arithmetic Series A series is called an arithmetic series if; d  Tn  Tn1 where d is a constant, called the common difference The general term of an arithmetic series with first term a and common difference d is; Tn  a  n  1d Three numbers a, x and b are terms in an arithmetic series if; bx  xa
  • 8. Applications of Arithmetic & Geometric Series Formulae for Arithmetic Series A series is called an arithmetic series if; d  Tn  Tn1 where d is a constant, called the common difference The general term of an arithmetic series with first term a and common difference d is; Tn  a  n  1d Three numbers a, x and b are terms in an arithmetic series if; ab bx  xa i.e. x  2
  • 9. The sum of the first n terms of an arithmetic series is;
  • 10. The sum of the first n terms of an arithmetic series is; n S n  a  l  (use when the last term l  Tn is known) 2
  • 11. The sum of the first n terms of an arithmetic series is; n S n  a  l  (use when the last term l  Tn is known) 2 n Sn  2a  n  1d  (use when the difference d is known) 2
  • 12. The sum of the first n terms of an arithmetic series is; n S n  a  l  (use when the last term l  Tn is known) 2 n Sn  2a  n  1d  (use when the difference d is known) 2 2007 HSC Question 3b) Heather decides to swim every day to improve her fitness level. On the first day she swims 750 metres, and on each day after that she swims 100 metres more than the previous day. That is she swims 850 metres on the second day, 950 metres on the third day and so on.
  • 13. The sum of the first n terms of an arithmetic series is; n S n  a  l  (use when the last term l  Tn is known) 2 n Sn  2a  n  1d  (use when the difference d is known) 2 2007 HSC Question 3b) Heather decides to swim every day to improve her fitness level. On the first day she swims 750 metres, and on each day after that she swims 100 metres more than the previous day. That is she swims 850 metres on the second day, 950 metres on the third day and so on. (i) Write down a formula for the distance she swims on he nth day.
  • 14. The sum of the first n terms of an arithmetic series is; n S n  a  l  (use when the last term l  Tn is known) 2 n Sn  2a  n  1d  (use when the difference d is known) 2 2007 HSC Question 3b) Heather decides to swim every day to improve her fitness level. On the first day she swims 750 metres, and on each day after that she swims 100 metres more than the previous day. That is she swims 850 metres on the second day, 950 metres on the third day and so on. (i) Write down a formula for the distance she swims on he nth day. a  750
  • 15. The sum of the first n terms of an arithmetic series is; n S n  a  l  (use when the last term l  Tn is known) 2 n Sn  2a  n  1d  (use when the difference d is known) 2 2007 HSC Question 3b) Heather decides to swim every day to improve her fitness level. On the first day she swims 750 metres, and on each day after that she swims 100 metres more than the previous day. That is she swims 850 metres on the second day, 950 metres on the third day and so on. (i) Write down a formula for the distance she swims on he nth day. a  750 d  100
  • 16. The sum of the first n terms of an arithmetic series is; n S n  a  l  (use when the last term l  Tn is known) 2 n Sn  2a  n  1d  (use when the difference d is known) 2 2007 HSC Question 3b) Heather decides to swim every day to improve her fitness level. On the first day she swims 750 metres, and on each day after that she swims 100 metres more than the previous day. That is she swims 850 metres on the second day, 950 metres on the third day and so on. (i) Write down a formula for the distance she swims on he nth day. a  750 Find a particular term, Tn d  100
  • 17. The sum of the first n terms of an arithmetic series is; n S n  a  l  (use when the last term l  Tn is known) 2 n Sn  2a  n  1d  (use when the difference d is known) 2 2007 HSC Question 3b) Heather decides to swim every day to improve her fitness level. On the first day she swims 750 metres, and on each day after that she swims 100 metres more than the previous day. That is she swims 850 metres on the second day, 950 metres on the third day and so on. (i) Write down a formula for the distance she swims on he nth day. a  750 Find a particular term, Tn d  100 Tn  a  n  1d
  • 18. The sum of the first n terms of an arithmetic series is; n S n  a  l  (use when the last term l  Tn is known) 2 n Sn  2a  n  1d  (use when the difference d is known) 2 2007 HSC Question 3b) Heather decides to swim every day to improve her fitness level. On the first day she swims 750 metres, and on each day after that she swims 100 metres more than the previous day. That is she swims 850 metres on the second day, 950 metres on the third day and so on. (i) Write down a formula for the distance she swims on he nth day. a  750 Find a particular term, Tn Tn  750   n  1100 d  100 Tn  a  n  1d
  • 19. The sum of the first n terms of an arithmetic series is; n S n  a  l  (use when the last term l  Tn is known) 2 n Sn  2a  n  1d  (use when the difference d is known) 2 2007 HSC Question 3b) Heather decides to swim every day to improve her fitness level. On the first day she swims 750 metres, and on each day after that she swims 100 metres more than the previous day. That is she swims 850 metres on the second day, 950 metres on the third day and so on. (i) Write down a formula for the distance she swims on he nth day. a  750 Find a particular term, Tn Tn  750   n  1100 d  100 Tn  a  n  1d Tn  650  100n
  • 20. (ii) How far does she swim on the 10th day? (1)
  • 21. (ii) How far does she swim on the 10th day? (1) Find a particular term, T10
  • 22. (ii) How far does she swim on the 10th day? (1) Find a particular term, T10 Tn  650  100n
  • 23. (ii) How far does she swim on the 10th day? (1) Find a particular term, T10 Tn  650  100n T10  650  100(10) T10  1650
  • 24. (ii) How far does she swim on the 10th day? (1) Find a particular term, T10 Tn  650  100n T10  650  100(10) T10  1650  she swims 1650 metres on the 10th day
  • 25. (ii) How far does she swim on the 10th day? (1) Find a particular term, T10 Tn  650  100n T10  650  100(10) T10  1650  she swims 1650 metres on the 10th day (iii) What is the total distance she swims in the first 10 days? (1)
  • 26. (ii) How far does she swim on the 10th day? (1) Find a particular term, T10 Tn  650  100n T10  650  100(10) T10  1650  she swims 1650 metres on the 10th day (iii) What is the total distance she swims in the first 10 days? (1) Find a total amount, S10
  • 27. (ii) How far does she swim on the 10th day? (1) Find a particular term, T10 Tn  650  100n T10  650  100(10) T10  1650  she swims 1650 metres on the 10th day (iii) What is the total distance she swims in the first 10 days? (1) Find a total amount, S10 n S n  2a  n  1d  2
  • 28. (ii) How far does she swim on the 10th day? (1) Find a particular term, T10 Tn  650  100n T10  650  100(10) T10  1650  she swims 1650 metres on the 10th day (iii) What is the total distance she swims in the first 10 days? (1) Find a total amount, S10 n S n  2a  n  1d  2 10 S10  2(750)  9(100) 2 S10  5 1500  900   12000
  • 29. (ii) How far does she swim on the 10th day? (1) Find a particular term, T10 Tn  650  100n T10  650  100(10) T10  1650  she swims 1650 metres on the 10th day (iii) What is the total distance she swims in the first 10 days? (1) Find a total amount, S10 n n S n  2a  n  1d  Sn  a  l  2 OR 2 10 S10  2(750)  9(100) 2 S10  5 1500  900   12000
  • 30. (ii) How far does she swim on the 10th day? (1) Find a particular term, T10 Tn  650  100n T10  650  100(10) T10  1650  she swims 1650 metres on the 10th day (iii) What is the total distance she swims in the first 10 days? (1) Find a total amount, S10 n n S n  2a  n  1d  Sn  a  l  2 OR 2 10 10 S10  2(750)  9(100) S10   750  1650  2 2 S10  5 1500  900   12000  12000
  • 31. (ii) How far does she swim on the 10th day? (1) Find a particular term, T10 Tn  650  100n T10  650  100(10) T10  1650  she swims 1650 metres on the 10th day (iii) What is the total distance she swims in the first 10 days? (1) Find a total amount, S10 n n S n  2a  n  1d  Sn  a  l  2 OR 2 10 10 S10  2(750)  9(100) S10   750  1650  2 2 S10  5 1500  900   12000  12000  she swims a total of 12000 metres in 10 days
  • 32. (iv) After how many days does the total distance she has swum equal (2) the width of the English Channel, a distance of 34 kilometres?
  • 33. (iv) After how many days does the total distance she has swum equal (2) the width of the English Channel, a distance of 34 kilometres? Find a total amount, Sn
  • 34. (iv) After how many days does the total distance she has swum equal (2) the width of the English Channel, a distance of 34 kilometres? Find a total amount, Sn n S n  2a  n  1d  2
  • 35. (iv) After how many days does the total distance she has swum equal (2) the width of the English Channel, a distance of 34 kilometres? Find a total amount, Sn n S n  2a  n  1d  2 n 34000  1500   n  1100 2
  • 36. (iv) After how many days does the total distance she has swum equal (2) the width of the English Channel, a distance of 34 kilometres? Find a total amount, Sn n S n  2a  n  1d  2 n 34000  1500   n  1100 2 68000  n 1400  100n  68000  1400n  100n 2 100n 2  1400n  68000  0 n 2  14n  680  0
  • 37. (iv) After how many days does the total distance she has swum equal (2) the width of the English Channel, a distance of 34 kilometres? Find a total amount, Sn n S n  2a  n  1d  2 n 34000  1500   n  1100 2 68000  n 1400  100n  68000  1400n  100n 2 100n 2  1400n  68000  0 n 2  14n  680  0  n  34  n  20   0
  • 38. (iv) After how many days does the total distance she has swum equal (2) the width of the English Channel, a distance of 34 kilometres? Find a total amount, Sn n S n  2a  n  1d  2 n 34000  1500   n  1100 2 68000  n 1400  100n  68000  1400n  100n 2 100n 2  1400n  68000  0 n 2  14n  680  0  n  34  n  20   0  n  34 or n  20
  • 39. (iv) After how many days does the total distance she has swum equal (2) the width of the English Channel, a distance of 34 kilometres? Find a total amount, Sn n S n  2a  n  1d  2 n 34000  1500   n  1100 2 68000  n 1400  100n  68000  1400n  100n 2 100n 2  1400n  68000  0 n 2  14n  680  0  n  34  n  20   0  n  34 or n  20  it takes 20 days to swim 34 kilometres
  • 40. Formulae for Geometric Series
  • 41. Formulae for Geometric Series A series is called a geometric series if;
  • 42. Formulae for Geometric Series A series is called a geometric series if; Tn r Tn1 where r is a constant, called the common ratio
  • 43. Formulae for Geometric Series A series is called a geometric series if; Tn r Tn1 where r is a constant, called the common ratio The general term of a geometric series with first term a and common ratio r is;
  • 44. Formulae for Geometric Series A series is called a geometric series if; Tn r Tn1 where r is a constant, called the common ratio The general term of a geometric series with first term a and common ratio r is; Tn  ar n1
  • 45. Formulae for Geometric Series A series is called a geometric series if; Tn r Tn1 where r is a constant, called the common ratio The general term of a geometric series with first term a and common ratio r is; Tn  ar n1 Three numbers a, x and b are terms in a geometric series if; b x  x a
  • 46. Formulae for Geometric Series A series is called a geometric series if; Tn r Tn1 where r is a constant, called the common ratio The general term of a geometric series with first term a and common ratio r is; Tn  ar n1 Three numbers a, x and b are terms in a geometric series if; b x  i.e. x   ab x a
  • 47. Formulae for Geometric Series A series is called a geometric series if; Tn r Tn1 where r is a constant, called the common ratio The general term of a geometric series with first term a and common ratio r is; Tn  ar n1 Three numbers a, x and b are terms in a geometric series if; b x  i.e. x   ab x a The sum of the first n terms of a geometric series is;
  • 48. Formulae for Geometric Series A series is called a geometric series if; Tn r Tn1 where r is a constant, called the common ratio The general term of a geometric series with first term a and common ratio r is; Tn  ar n1 Three numbers a, x and b are terms in a geometric series if; b x  i.e. x   ab x a The sum of the first n terms of a geometric series is; a  r n  1 Sn  (easier when r  1) r 1
  • 49. Formulae for Geometric Series A series is called a geometric series if; Tn r Tn1 where r is a constant, called the common ratio The general term of a geometric series with first term a and common ratio r is; Tn  ar n1 Three numbers a, x and b are terms in a geometric series if; b x  i.e. x   ab x a The sum of the first n terms of a geometric series is; a  r n  1 Sn  (easier when r  1) r 1 a 1  r n  Sn  (easier when r  1) 1 r
  • 50. e.g. A company’s sales are declining by 6% every year, with 50000 items sold in 2001. During which year will sales first fall below 20000?
  • 51. e.g. A company’s sales are declining by 6% every year, with 50000 items sold in 2001. During which year will sales first fall below 20000? a  50000
  • 52. e.g. A company’s sales are declining by 6% every year, with 50000 items sold in 2001. During which year will sales first fall below 20000? a  50000 r  0.94
  • 53. e.g. A company’s sales are declining by 6% every year, with 50000 items sold in 2001. During which year will sales first fall below 20000? a  50000 r  0.94 Tn  20000
  • 54. e.g. A company’s sales are declining by 6% every year, with 50000 items sold in 2001. During which year will sales first fall below 20000? a  50000 Tn  ar n1 r  0.94 Tn  20000
  • 55. e.g. A company’s sales are declining by 6% every year, with 50000 items sold in 2001. During which year will sales first fall below 20000? a  50000 Tn  ar n1 20000  50000  0.94  n1 r  0.94 Tn  20000
  • 56. e.g. A company’s sales are declining by 6% every year, with 50000 items sold in 2001. During which year will sales first fall below 20000? a  50000 Tn  ar n1 20000  50000  0.94  n1 r  0.94 0.4   0.94  n1 Tn  20000  0.94   0.4 n1
  • 57. e.g. A company’s sales are declining by 6% every year, with 50000 items sold in 2001. During which year will sales first fall below 20000? a  50000 Tn  ar n1 20000  50000  0.94  n1 r  0.94 0.4   0.94  n1 Tn  20000  0.94   0.4 n1 log  0.94   log 0.4 n1
  • 58. e.g. A company’s sales are declining by 6% every year, with 50000 items sold in 2001. During which year will sales first fall below 20000? a  50000 Tn  ar n1 20000  50000  0.94  n1 r  0.94 0.4   0.94  n1 Tn  20000  0.94   0.4 n1 log  0.94   log 0.4 n1  n  1 log  0.94   log 0.4
  • 59. e.g. A company’s sales are declining by 6% every year, with 50000 items sold in 2001. During which year will sales first fall below 20000? a  50000 Tn  ar n1 20000  50000  0.94  n1 r  0.94 0.4   0.94  n1 Tn  20000  0.94   0.4 n1 log  0.94   log 0.4 n1  n  1 log  0.94   log 0.4 log 0.4  n  1  log 0.94
  • 60. e.g. A company’s sales are declining by 6% every year, with 50000 items sold in 2001. During which year will sales first fall below 20000? a  50000 Tn  ar n1 20000  50000  0.94  n1 r  0.94 0.4   0.94  n1 Tn  20000  0.94   0.4 n1 log  0.94   log 0.4 n1  n  1 log  0.94   log 0.4 log 0.4  n  1  log 0.94  n  1  14.80864248 n  15.80864248
  • 61. e.g. A company’s sales are declining by 6% every year, with 50000 items sold in 2001. During which year will sales first fall below 20000? a  50000 Tn  ar n1 20000  50000  0.94  n1 r  0.94 0.4   0.94  n1 Tn  20000  0.94   0.4 n1 log  0.94   log 0.4 n1  n  1 log  0.94   log 0.4 log 0.4  n  1  log 0.94  n  1  14.80864248 n  15.80864248  during the 16th year (i.e. 2016) sales will fall below 20000
  • 62. 2005 HSC Question 7a) Anne and Kay are employed by an accounting firm. Anne accepts employment with an initial annual salary of $50000. In each of the following years her annual salary is increased by $2500. Kay accepts employment with an initial annual salary of $50000. in each of the following years her annual salary is increased by 4%
  • 63. 2005 HSC Question 7a) Anne and Kay are employed by an accounting firm. Anne accepts employment with an initial annual salary of $50000. In each of the following years her annual salary is increased by $2500. Kay accepts employment with an initial annual salary of $50000. in each of the following years her annual salary is increased by 4% arithmetic
  • 64. 2005 HSC Question 7a) Anne and Kay are employed by an accounting firm. Anne accepts employment with an initial annual salary of $50000. In each of the following years her annual salary is increased by $2500. Kay accepts employment with an initial annual salary of $50000. in each of the following years her annual salary is increased by 4% arithmetic a = 50000 d = 2500
  • 65. 2005 HSC Question 7a) Anne and Kay are employed by an accounting firm. Anne accepts employment with an initial annual salary of $50000. In each of the following years her annual salary is increased by $2500. Kay accepts employment with an initial annual salary of $50000. in each of the following years her annual salary is increased by 4% arithmetic geometric a = 50000 d = 2500 a = 50000 r = 1.04
  • 66. 2005 HSC Question 7a) Anne and Kay are employed by an accounting firm. Anne accepts employment with an initial annual salary of $50000. In each of the following years her annual salary is increased by $2500. Kay accepts employment with an initial annual salary of $50000. in each of the following years her annual salary is increased by 4% arithmetic geometric a = 50000 d = 2500 a = 50000 r = 1.04 (i) What is Anne’s salary in her thirteenth year? (2)
  • 67. 2005 HSC Question 7a) Anne and Kay are employed by an accounting firm. Anne accepts employment with an initial annual salary of $50000. In each of the following years her annual salary is increased by $2500. Kay accepts employment with an initial annual salary of $50000. in each of the following years her annual salary is increased by 4% arithmetic geometric a = 50000 d = 2500 a = 50000 r = 1.04 (i) What is Anne’s salary in her thirteenth year? (2) Find a particular term, T13
  • 68. 2005 HSC Question 7a) Anne and Kay are employed by an accounting firm. Anne accepts employment with an initial annual salary of $50000. In each of the following years her annual salary is increased by $2500. Kay accepts employment with an initial annual salary of $50000. in each of the following years her annual salary is increased by 4% arithmetic geometric a = 50000 d = 2500 a = 50000 r = 1.04 (i) What is Anne’s salary in her thirteenth year? (2) Find a particular term, T13 Tn  a  n  1d
  • 69. 2005 HSC Question 7a) Anne and Kay are employed by an accounting firm. Anne accepts employment with an initial annual salary of $50000. In each of the following years her annual salary is increased by $2500. Kay accepts employment with an initial annual salary of $50000. in each of the following years her annual salary is increased by 4% arithmetic geometric a = 50000 d = 2500 a = 50000 r = 1.04 (i) What is Anne’s salary in her thirteenth year? (2) Find a particular term, T13 Tn  a  n  1d T13  50000  12  2500  T13  80000
  • 70. 2005 HSC Question 7a) Anne and Kay are employed by an accounting firm. Anne accepts employment with an initial annual salary of $50000. In each of the following years her annual salary is increased by $2500. Kay accepts employment with an initial annual salary of $50000. in each of the following years her annual salary is increased by 4% arithmetic geometric a = 50000 d = 2500 a = 50000 r = 1.04 (i) What is Anne’s salary in her thirteenth year? (2) Find a particular term, T13 Tn  a  n  1d T13  50000  12  2500  T13  80000  Anne earns $80000 in her 13th year
  • 71. (ii) What is Kay’s annual salary in her thirteenth year? (2)
  • 72. (ii) What is Kay’s annual salary in her thirteenth year? (2) Find a particular term, T13
  • 73. (ii) What is Kay’s annual salary in her thirteenth year? (2) Find a particular term, T13 Tn  ar n1
  • 74. (ii) What is Kay’s annual salary in her thirteenth year? (2) Find a particular term, T13 Tn  ar n1 T13  50000 1.04  12 T13  80051.61093
  • 75. (ii) What is Kay’s annual salary in her thirteenth year? (2) Find a particular term, T13 Tn  ar n1 T13  50000 1.04  12 T13  80051.61093  Kay earns $80051.61 in her thirteenth year
  • 76. (ii) What is Kay’s annual salary in her thirteenth year? (2) Find a particular term, T13 Tn  ar n1 T13  50000 1.04  12 T13  80051.61093  Kay earns $80051.61 in her thirteenth year (iii) By what amount does the total amount paid to Kay in her first (3) twenty years exceed that paid to Anne in her first twenty years?
  • 77. (ii) What is Kay’s annual salary in her thirteenth year? (2) Find a particular term, T13 Tn  ar n1 T13  50000 1.04  12 T13  80051.61093  Kay earns $80051.61 in her thirteenth year (iii) By what amount does the total amount paid to Kay in her first (3) twenty years exceed that paid to Anne in her first twenty years? Find a total amount, S20
  • 78. (ii) What is Kay’s annual salary in her thirteenth year? (2) Find a particular term, T13 Tn  ar n1 T13  50000 1.04  12 T13  80051.61093  Kay earns $80051.61 in her thirteenth year (iii) By what amount does the total amount paid to Kay in her first (3) twenty years exceed that paid to Anne in her first twenty years? Find a total amount, S20 n Anne S n  2a  n  1d  2
  • 79. (ii) What is Kay’s annual salary in her thirteenth year? (2) Find a particular term, T13 Tn  ar n1 T13  50000 1.04  12 T13  80051.61093  Kay earns $80051.61 in her thirteenth year (iii) By what amount does the total amount paid to Kay in her first (3) twenty years exceed that paid to Anne in her first twenty years? Find a total amount, S20 n Anne S n  2a  n  1d  2 20 S20  2(50000)  19(2500) 2 S10  10 100000  47500   1475000
  • 80. (ii) What is Kay’s annual salary in her thirteenth year? (2) Find a particular term, T13 Tn  ar n1 T13  50000 1.04  12 T13  80051.61093  Kay earns $80051.61 in her thirteenth year (iii) By what amount does the total amount paid to Kay in her first (3) twenty years exceed that paid to Anne in her first twenty years? Find a total amount, S20 a  r n  1 n Anne S n  2a  n  1d  Kay S n  2 n 1 20 S20  2(50000)  19(2500) 2 S10  10 100000  47500   1475000
  • 81. (ii) What is Kay’s annual salary in her thirteenth year? (2) Find a particular term, T13 Tn  ar n1 T13  50000 1.04  12 T13  80051.61093  Kay earns $80051.61 in her thirteenth year (iii) By what amount does the total amount paid to Kay in her first (3) twenty years exceed that paid to Anne in her first twenty years? Find a total amount, S20 a  r n  1 n Anne S n  2a  n  1d  Kay S n  2 n 1 20 S20  2(50000)  19(2500) 50000 1.0420  1 2 S20  0.04 S10  10 100000  47500   1488903.93  1475000
  • 82. (ii) What is Kay’s annual salary in her thirteenth year? (2) Find a particular term, T13 Tn  ar n1 T13  50000 1.04  12 T13  80051.61093  Kay earns $80051.61 in her thirteenth year (iii) By what amount does the total amount paid to Kay in her first (3) twenty years exceed that paid to Anne in her first twenty years? Find a total amount, S20 a  r n  1 n Anne S n  2a  n  1d  Kay S n  2 n 1 20 S20  2(50000)  19(2500) 50000 1.0420  1 2 S20  0.04 S10  10 100000  47500   1488903.93  1475000  Kay is paid $13903.93 more than Anne
  • 83. The limiting sum S exists if and only if  1  r  1, in which case;
  • 84. The limiting sum S exists if and only if  1  r  1, in which case; a S  1 r
  • 85. The limiting sum S exists if and only if  1  r  1, in which case; a 2007 HSC Question 1d) S  (2) Find the limiting sum of the geometric series 1 r 3 3 3    4 16 64
  • 86. The limiting sum S exists if and only if  1  r  1, in which case; a 2007 HSC Question 1d) S  (2) Find the limiting sum of the geometric series 1 r 3 3 3    4 16 64 3 4 3 r  a 16 3 4 1  4
  • 87. The limiting sum S exists if and only if  1  r  1, in which case; a 2007 HSC Question 1d) S  (2) Find the limiting sum of the geometric series 1 r 3 3 3 a    S  4 16 64 1 r 3 4 3 r  a 16 3 4 1  4
  • 88. The limiting sum S exists if and only if  1  r  1, in which case; a 2007 HSC Question 1d) S  (2) Find the limiting sum of the geometric series 1 r 3 3 3 a    S  4 16 64 1 r 3 4 3 3 r  a 16 3 S  4 1 4 1 1  4 4 1
  • 89. The limiting sum S exists if and only if  1  r  1, in which case; a 2007 HSC Question 1d) S  (2) Find the limiting sum of the geometric series 1 r 3 3 3 a    S  4 16 64 1 r 3 4 3 3 r  a 16 3 S  4 1 4 1 1  4 4 1 2003 HSC Question 7a) (i) Find the limiting sum of the geometric series (2) 2 2 2   2 1   2 2 1
  • 90. The limiting sum S exists if and only if  1  r  1, in which case; a 2007 HSC Question 1d) S  (2) Find the limiting sum of the geometric series 1 r 3 3 3 a    S  4 16 64 1 r 3 4 3 3 r  a 16 3 S  4 1 4 1 1  4 4 1 2003 HSC Question 7a) (i) Find the limiting sum of the geometric series (2) 2 2 2   2 1   2 2 1 2 1 r  2 1 2 a2 1  2 1
  • 91. The limiting sum S exists if and only if  1  r  1, in which case; a 2007 HSC Question 1d) S  (2) Find the limiting sum of the geometric series 1 r 3 3 3 a    S  4 16 64 1 r 3 4 3 3 r  a 16 3 S  4 1 4 1 1  4 4 1 2003 HSC Question 7a) (i) Find the limiting sum of the geometric series a (2) S  2 2  2  1 r 2 1   2 2 1 2 1 r  2 1 2 a2 1  2 1
  • 92. The limiting sum S exists if and only if  1  r  1, in which case; a 2007 HSC Question 1d) S  (2) Find the limiting sum of the geometric series 1 r 3 3 3 a    S  4 16 64 1 r 3 4 3 3 r  a 16 3 S  4 1 4 1 1  4 4 1 2003 HSC Question 7a) (i) Find the limiting sum of the geometric series a (2) S  2 2  2  1 r 2 1   2 2 1 2 1 S  2 r  1 2 1 2 1 2 1 a2 1  2 1
  • 93. 2 2 1 S   1 2  1 1
  • 94. 2 2 1 S   1 2  1 1 2 2 1   1 2  2   2 1  2 2
  • 95. 2 2 1 S   1 2 1 1 2 2 1   1 2  2  2 1   2 2 (ii) Explain why the geometric series 2 2 2    does NOT have a limiting sum. (1) 2 1   2 2 1
  • 96. 2 2 1 S   1 2 1 1 2 2 1   1 2  2  2 1   2 2 (ii) Explain why the geometric series 2 2 2    does NOT have a limiting sum. (1) 2 1   2 2 1 Limiting sums only occur when – 1< r< 1
  • 97. 2 2 1 S   1 2 1 1 2 2 1   1 2  2  2 1   2 2 (ii) Explain why the geometric series 2 2 2    does NOT have a limiting sum. (1) 2 1   2 2 1 Limiting sums only occur when – 1< r< 1 2 1 r  2 1 2 1  2 1
  • 98. 2 2 1 S   1 2 1 1 2 2 1   1 2  2  2 1   2 2 (ii) Explain why the geometric series 2 2 2    does NOT have a limiting sum. (1) 2 1   2 2 1 Limiting sums only occur when – 1< r< 1 2 1 r   r  2.414  1 2 1 2 1  2 1
  • 99. 2 2 1 S   1 2 1 1 2 2 1   1 2  2  2 1   2 2 (ii) Explain why the geometric series 2 2 2    does NOT have a limiting sum. (1) 2 1   2 2 1 Limiting sums only occur when – 1< r< 1 2 1 r   r  2.414  1 2 1 2 1 as r >1, no limiting sum exists  2 1
  • 100. 2004 HSC Question 9a) Consider the series 1  tan 2   tan 4   
  • 101. 2004 HSC Question 9a) Consider the series 1  tan 2   tan 4    a 1 r   tan 2 
  • 102. 2004 HSC Question 9a) Consider the series 1  tan 2   tan 4    a 1 r   tan 2  (i) When the limiting sum exists, find its value in simplest form. (2)
  • 103. 2004 HSC Question 9a) Consider the series 1  tan 2   tan 4    a 1 r   tan 2  (i) When the limiting sum exists, find its value in simplest form. (2) a S  1 r
  • 104. 2004 HSC Question 9a) Consider the series 1  tan 2   tan 4    a 1 r   tan 2  (i) When the limiting sum exists, find its value in simplest form. (2) a S  1 r 1 S  1  tan 2 
  • 105. 2004 HSC Question 9a) Consider the series 1  tan 2   tan 4    a 1 r   tan 2  (i) When the limiting sum exists, find its value in simplest form. (2) a S  1 r 1 S  1  tan 2  1  sec 2   cos 2 
  • 106.   (ii) For what values of  in the interval    (2) 2 2 does the limiting sum of the series exist?
  • 107.   (ii) For what values of  in the interval    (2) 2 2 does the limiting sum of the series exist? Limiting sums only occur when – 1< r< 1
  • 108.   (ii) For what values of  in the interval    (2) 2 2 does the limiting sum of the series exist? Limiting sums only occur when – 1< r< 1 1   tan 2   1
  • 109.   (ii) For what values of  in the interval    (2) 2 2 does the limiting sum of the series exist? Limiting sums only occur when – 1< r< 1 1   tan 2   1 1  tan 2   1 1  tan 2   1
  • 110.   (ii) For what values of  in the interval    (2) 2 2 does the limiting sum of the series exist? Limiting sums only occur when – 1< r< 1 1   tan 2   1 1  tan 2   1 1  tan 2   1 1  tan   1
  • 111.   (ii) For what values of  in the interval    (2) 2 2 does the limiting sum of the series exist? Limiting sums only occur when – 1< r< 1 1   tan 2   1 1  tan 2   1 1  tan 2   1 1  tan   1  Now tan   1, when    4
  • 112.   (ii) For what values of  in the interval    (2) 2 2 does the limiting sum of the series exist? Limiting sums only occur when – 1< r< 1 1   tan 2   1 1  tan 2   1 1  tan 2   1 1  tan   1  Now tan   1, when    y 4 y  tan x 1   x  2 –1 2
  • 113.   (ii) For what values of  in the interval    (2) 2 2 does the limiting sum of the series exist? Limiting sums only occur when – 1< r< 1 1   tan 2   1 1  tan 2   1 1  tan 2   1 1  tan   1  Now tan   1, when    y 4 y  tan x 1   x    2 –1 2    4 4
  • 114.   (ii) For what values of  in the interval    (2) 2 2 does the limiting sum of the series exist? Limiting sums only occur when – 1< r< 1 1   tan 2   1 1  tan 2   1 1  tan 2   1 1  tan   1 Exercise 7A;  3, 4, 5, 10, 11, Now tan   1, when    4 16, 17, 20* y y  tan x 1   x    2 –1 2    4 4