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 SeriesA series is called an arithmetic series if;d Tn Tn1where d is a constant, called the common difference 4. Applications of Arithmetic &Geometric Series Formulae for Arithmetic SeriesA series is called an arithmetic series if;d Tn Tn1where d is a constant, called the common differenceThe general term of an arithmetic series with first term a and commondifference d is; 5. Applications of Arithmetic &Geometric Series Formulae for Arithmetic SeriesA series is called an arithmetic series if;d Tn Tn1where d is a constant, called the common differenceThe general term of an arithmetic series with first term a and commondifference d is; Tn a n 1d 6. Applications of Arithmetic &Geometric Series Formulae for Arithmetic SeriesA series is called an arithmetic series if;d Tn Tn1where d is a constant, called the common differenceThe general term of an arithmetic series with first term a and commondifference d is; Tn a n 1dThree numbers a, x and b are terms in an arithmetic series if; 7. Applications of Arithmetic &Geometric Series Formulae for Arithmetic SeriesA series is called an arithmetic series if;d Tn Tn1where d is a constant, called the common differenceThe general term of an arithmetic series with first term a and commondifference d is; Tn a n 1dThree numbers a, x and b are terms in an arithmetic series if;bx xa 8. Applications of Arithmetic &Geometric Series Formulae for Arithmetic SeriesA series is called an arithmetic series if;d Tn Tn1where d is a constant, called the common differenceThe general term of an arithmetic series with first term a and commondifference d is; Tn a n 1dThree numbers a, x and b are terms in an arithmetic series if;ab bx xai.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)2n Sn 2a n 1d (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)2n Sn 2a n 1d (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)2n Sn 2a n 1d (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)2n Sn 2a n 1d (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)2n Sn 2a n 1d (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)2n Sn 2a n 1d (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)2n Sn 2a n 1d (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 1d 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)2n Sn 2a n 1d (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 1100 d 100 Tn a n 1d 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)2n Sn 2a n 1d (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 1100 d 100 Tn a n 1d 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, T10Tn 650 100n 23. (ii) How far does she swim on the 10th day? (1)Find a particular term, T10Tn 650 100n T10 650 100(10) T10 1650 24. (ii) How far does she swim on the 10th day? (1)Find a particular term, T10Tn 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, T10Tn 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, T10Tn 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, T10Tn 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 nS n 2a n 1d 2 28. (ii) How far does she swim on the 10th day?(1)Find a particular term, T10Tn 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 nS n 2a n 1d 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, T10Tn 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 nnS n 2a n 1d 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, T10Tn 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 nS n 2a n 1d Sn a l 2 OR2 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, T10Tn 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 nS n 2a n 1d Sn a l 2 OR2 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 1d 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 1d 2n34000 1500 n 11002 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 1d 2n34000 1500 n 1100268000 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 1d 2n34000 1500 n 1100268000 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 1d 2n34000 1500 n 1100268000 n 1400 100n 68000 1400n 100n 2 100n 2 1400n 68000 0 n 2 14n 680 0 n 34 n 20 0 n 34orn 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 1d 2n34000 1500 n 1100268000 n 1400 100n 68000 1400n 100n 2 100n 2 1400n 68000 0 n 2 14n 680 0 n 34 n 20 0 n 34orn 20 it takes 20 days to swim 34 kilometres 40. Formulae for Geometric Series 41. Formulae for Geometric SeriesA series is called a geometric series if; 42. Formulae for Geometric SeriesA series is called a geometric series if; TnrTn1where r is a constant, called the common ratio 43. Formulae for Geometric SeriesA series is called a geometric series if; TnrTn1where r is a constant, called the common ratioThe general term of a geometric series with first term a and commonratio r is; 44. Formulae for Geometric SeriesA series is called a geometric series if; TnrTn1where r is a constant, called the common ratioThe general term of a geometric series with first term a and commonratio r is;Tn ar n1 45. Formulae for Geometric SeriesA series is called a geometric series if; TnrTn1where r is a constant, called the common ratioThe general term of a geometric series with first term a and commonratio r is;Tn ar n1Three numbers a, x and b are terms in a geometric series if;b xx a 46. Formulae for Geometric SeriesA series is called a geometric series if; TnrTn1where r is a constant, called the common ratioThe general term of a geometric series with first term a and commonratio r is;Tn ar n1Three numbers a, x and b are terms in a geometric series if;b x i.e. x abx a 47. Formulae for Geometric SeriesA series is called a geometric series if; TnrTn1where r is a constant, called the common ratioThe general term of a geometric series with first term a and commonratio r is;Tn ar n1Three numbers a, x and b are terms in a geometric series if;b x i.e. x abx aThe sum of the first n terms of a geometric series is; 48. Formulae for Geometric SeriesA series is called a geometric series if; TnrTn1where r is a constant, called the common ratioThe general term of a geometric series with first term a and commonratio r is;Tn ar n1Three numbers a, x and b are terms in a geometric series if;b x i.e. x abx aThe sum of the first n terms of a geometric series is;a r n 1Sn (easier when r 1) r 1 49. Formulae for Geometric SeriesA series is called a geometric series if; TnrTn1where r is a constant, called the common ratioThe general term of a geometric series with first term a and commonratio r is;Tn ar n1Three numbers a, x and b are terms in a geometric series if;b x i.e. x abx aThe sum of the first n terms of a geometric series is;a r n 1Sn (easier when r 1) r 1 a 1 r n Sn (easier when r 1) 1 r 50. e.g. A companys sales are declining by 6% every year, with 50000items sold in 2001. During which year will sales first fall below 20000? 51. e.g. A companys sales are declining by 6% every year, with 50000items sold in 2001. During which year will sales first fall below 20000? a 50000 52. e.g. A companys sales are declining by 6% every year, with 50000items sold in 2001. During which year will sales first fall below 20000? a 50000 r 0.94 53. e.g. A companys sales are declining by 6% every year, with 50000items sold in 2001. During which year will sales first fall below 20000? a 50000 r 0.94Tn 20000 54. e.g. A companys sales are declining by 6% every year, with 50000items sold in 2001. During which year will sales first fall below 20000? a 50000Tn ar n1 r 0.94Tn 20000 55. e.g. A companys sales are declining by 6% every year, with 50000items sold in 2001. During which year will sales first fall below 20000? a 50000 Tn ar n120000 50000 0.94 n1 r 0.94Tn 20000 56. e.g. A companys sales are declining by 6% every year, with 50000items sold in 2001. During which year will sales first fall below 20000? a 50000 Tn ar n120000 50000 0...