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- 1. Applications of Arithmetic & Geometric Series 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.
- 2. Applications of Arithmetic & Geometric Series 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
- 3. Applications of Arithmetic & Geometric Series 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
- 4. Applications of Arithmetic & Geometric Series 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 1100 Tn 650 100n d 100 Tn a n 1d
- 5. (ii) How far does she swim on the 10th day? (1)
- 6. (ii) How far does she swim on the 10th day? Find a particular term, T10 (1)
- 7. (ii) How far does she swim on the 10th day? Find a particular term, T10 Tn 650 100n T10 650 100(10) T10 1650 (1)
- 8. (ii) How far does she swim on the 10th day? Find a particular term, T10 Tn 650 100n T10 650 100(10) T10 1650 she swims 1650 metres on the 10th day (1)
- 9. (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)
- 10. (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? Find a total amount, S10 (1)
- 11. (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? Find a total amount, S10 n S n 2a n 1d 2 10 S10 2(750) 9(100) 2 S10 5 1500 900 12000 (1)
- 12. (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? Find a total amount, S10 n S n 2a n 1d 2 10 S10 2(750) 9(100) 2 S10 5 1500 900 12000 n a l 2 10 S10 750 1650 2 12000 Sn OR (1)
- 13. (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? Find a total amount, S10 n S n 2a n 1d 2 10 S10 2(750) 9(100) 2 S10 5 1500 900 12000 n a l 2 10 S10 750 1650 2 12000 Sn OR (1) she swims a total of 12000 metres in 10 days
- 14. (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?
- 15. (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
- 16. (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 1d 2 n 34000 1500 n 1100 2
- 17. (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 1d 2 n 34000 1500 n 1100 2 68000 n 1400 100n 68000 1400n 100n 2 100n 2 1400n 68000 0
- 18. (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 1d 2 n 34000 1500 n 1100 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
- 19. (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 1d 2 n 34000 1500 n 1100 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
- 20. 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?
- 21. 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
- 22. 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 Tn ar n1 n1 20000 50000 0.94
- 23. 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 Tn ar n1 n1 20000 50000 0.94 0.4 0.94 n1 0.94 0.4 n1
- 24. 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 Tn ar n1 n1 20000 50000 0.94 0.4 0.94 n1 0.94 0.4 n1 log 0.94 log 0.4 n1
- 25. 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 Tn ar n1 n1 20000 50000 0.94 0.4 0.94 n1 0.94 0.4 n1 log 0.94 log 0.4 n 1 log 0.94 log 0.4 log 0.4 n 1 n1 log 0.94 n 1 14.80864248 n 15.80864248
- 26. 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 Tn ar n1 n1 20000 50000 0.94 0.4 0.94 n1 0.94 0.4 n1 log 0.94 log 0.4 n 1 log 0.94 log 0.4 log 0.4 n 1 n1 log 0.94 n 1 14.80864248 n 15.80864248 during the 16th year (i.e. 2016) sales will fall below 20000
- 27. 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%
- 28. 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
- 29. 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
- 30. 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)
- 31. 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
- 32. 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 1d T13 50000 12 2500 T13 80000 Anne earns $80000 in her 13th year
- 33. (ii) What is Kay’s annual salary in her thirteenth year? (2)
- 34. (ii) What is Kay’s annual salary in her thirteenth year? Find a particular term, T13 (2)
- 35. (ii) What is Kay’s annual salary in her thirteenth year? Find a particular term, T13 Tn ar n1 12 T13 50000 1.04 T13 80051.61093 Kay earns $80051.61 in her thirteenth year (2)
- 36. (ii) What is Kay’s annual salary in her thirteenth year? (2) Find a particular term, T13 Tn ar n1 12 T13 50000 1.04 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?
- 37. (ii) What is Kay’s annual salary in her thirteenth year? (2) Find a particular term, T13 Tn ar n1 12 T13 50000 1.04 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
- 38. (ii) What is Kay’s annual salary in her thirteenth year? (2) Find a particular term, T13 Tn ar n1 12 T13 50000 1.04 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 1d 2 20 S20 2(50000) 19(2500) 2 S10 10 100000 47500 1475000
- 39. (ii) What is Kay’s annual salary in her thirteenth year? (2) Find a particular term, T13 Tn ar n1 12 T13 50000 1.04 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 Kay Sn Anne S n 2a n 1d n 1 2 20 50000 1.0420 1 S20 2(50000) 19(2500) S 20 2 0.04 S10 10 100000 47500 1488903.93 1475000
- 40. (ii) What is Kay’s annual salary in her thirteenth year? (2) Find a particular term, T13 Tn ar n1 12 T13 50000 1.04 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 Kay Sn Anne S n 2a n 1d n 1 2 20 50000 1.0420 1 S20 2(50000) 19(2500) S 20 2 0.04 S10 10 100000 47500 1488903.93 1475000 Kay is paid $13903.93 more than Anne
- 41. 2007 HSC Question 1d) Find the limiting sum of the geometric series 3 3 3 4 16 64 (2)
- 42. 2007 HSC Question 1d) Find the limiting sum of the geometric series 3 3 3 4 16 64 a 3 4 3 4 r 16 3 1 4 (2)
- 43. (2) 2007 HSC Question 1d) Find the limiting sum of the geometric series 3 3 3 4 16 64 a 3 4 3 4 r 16 3 1 4 a S 1 r 3 S 4 1 1 4 1
- 44. (2) 2007 HSC Question 1d) Find the limiting sum of the geometric series 3 3 3 4 16 64 a 3 4 r 16 3 1 4 3 4 a S 1 r 3 S 4 1 1 4 1 2003 HSC Question 7a) (i) Find the limiting sum of the geometric series 2 2 2 2 2 1 2 1 (2)
- 45. (2) 2007 HSC Question 1d) Find the limiting sum of the geometric series 3 3 3 4 16 64 a 3 4 r 16 3 1 4 3 4 a S 1 r 3 S 4 1 1 4 1 2003 HSC Question 7a) (i) Find the limiting sum of the geometric series 2 2 2 2 2 1 2 1 2 1 r 2 1 2 1 a2 2 1 (2)
- 46. (2) 2007 HSC Question 1d) Find the limiting sum of the geometric series 3 3 3 4 16 64 a 3 4 r 16 3 1 4 3 4 a S 1 r 3 S 4 1 1 4 1 2003 HSC Question 7a) (i) Find the limiting sum of the geometric series 2 2 2 2 2 1 2 1 2 1 r 2 1 2 1 a2 2 1 (2) a S 1 r S 2 1 1 2 1
- 47. 2 2 1 S 1 2 1 1
- 48. 2 2 1 S 1 2 1 1 2 2 1 1 2 2 2 1 2 2
- 49. 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. 2 2 1 2 1 (1)
- 50. 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 (1) 2 does NOT have a limiting sum. 2 2 1 2 1 Limiting sums only occur when – 1< r< 1
- 51. 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 (1) 2 does NOT have a limiting sum. 2 2 1 2 1 Limiting sums only occur when – 1< r< 1 2 1 r r 2.414 1 2 1 2 1 2 1
- 52. 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 (1) 2 does NOT have a limiting sum. 2 2 1 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
- 53. 2004 HSC Question 9a) Consider the series 1 tan 2 tan 4
- 54. 2004 HSC Question 9a) Consider the series 1 tan 2 tan 4 a 1 r tan 2
- 55. 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)
- 56. 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. a S 1 r 1 S 1 tan 2 1 sec 2 cos 2 (2)
- 57. (ii) For what values of in the interval 2 does the limiting sum of the series exist? 2 (2)
- 58. (ii) For what values of in the interval 2 does the limiting sum of the series exist? 2 Limiting sums only occur when – 1< r< 1 (2)
- 59. (ii) For what values of in the interval 2 does the limiting sum of the series exist? 2 Limiting sums only occur when – 1< r< 1 1 tan 2 1 1 tan 2 1 1 tan 2 1 1 tan 1 (2)
- 60. (ii) For what values of in the interval 2 does the limiting sum of the series exist? 2 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 (2)
- 61. (ii) For what values of in the interval 2 does the limiting sum of the series exist? 2 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 y y tan x 1 2 –1 2 x (2)
- 62. (ii) For what values of in the interval 2 does the limiting sum of the series exist? (2) 2 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 y y tan x 1 2 2 –1 x 4 4
- 63. (ii) For what values of in the interval 2 does the limiting sum of the series exist? (2) 2 Limiting sums only occur when – 1< r< 1 1 tan 2 1 1 tan 2 1 1 tan 2 1 Exercise 7A; 1 tan 1 3, 4, 5, 10, 11, Now tan 1, when 16, 17, 20* 4 y y tan x 1 2 2 –1 x 4 4

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