11X1 T14 12 applications of ap & gp

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1. Applications of Arithmetic & Geometric SeriesFormulae for Arithmetic Series 2. Applications of Arithmetic & Geometric SeriesFormulae for Arithmetic Series A series is called an arithmetic series if; d Tn Tn1 where d is a constant, called the common difference 3. Applications of Arithmetic & Geometric SeriesFormulae for Arithmetic Series A series is called an arithmetic series if; d Tn Tn1 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 1d 4. Applications of Arithmetic & Geometric SeriesFormulae for Arithmetic Series A series is called an arithmetic series if; d Tn Tn1 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 1d Three numbers a, x and b are terms in an arithmetic series if;abbx xai.e. x 2 5. 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 6. 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 7. The sum of the first n terms of an arithmetic series is; nS n a l (use when the last term l Tn is known) 2 nSn 2a n 1d (use when the difference d is known) 22007 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 sheswims 100 metres more than the previous day.That is she swims 850 metres on the second day, 950 metres on the thirdday and so on. 8. The sum of the first n terms of an arithmetic series is; nS n a l (use when the last term l Tn is known) 2 nSn 2a n 1d (use when the difference d is known) 22007 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 sheswims 100 metres more than the previous day.That is she swims 850 metres on the second day, 950 metres on the thirdday and so on.(i) Write down a formula for the distance she swims on he nth day. 9. The sum of the first n terms of an arithmetic series is; nS n a l (use when the last term l Tn is known) 2 nSn 2a n 1d (use when the difference d is known) 22007 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 sheswims 100 metres more than the previous day.That is she swims 850 metres on the second day, 950 metres on the thirdday and so on.(i) Write down a formula for the distance she swims on he nth day.a 750 10. The sum of the first n terms of an arithmetic series is; nS n a l (use when the last term l Tn is known) 2 nSn 2a n 1d (use when the difference d is known) 22007 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 sheswims 100 metres more than the previous day.That is she swims 850 metres on the second day, 950 metres on the thirdday and so on.(i) Write down a formula for the distance she swims on he nth day.a 750d 100 11. The sum of the first n terms of an arithmetic series is; nS n a l (use when the last term l Tn is known) 2 nSn 2a n 1d (use when the difference d is known) 22007 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 sheswims 100 metres more than the previous day.That is she swims 850 metres on the second day, 950 metres on the thirdday and so on.(i) Write down a formula for the distance she swims on he nth day.a 750 Find a particular term, Tnd 100 12. The sum of the first n terms of an arithmetic series is; nS n a l (use when the last term l Tn is known) 2 nSn 2a n 1d (use when the difference d is known) 22007 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 sheswims 100 metres more than the previous day.That is she swims 850 metres on the second day, 950 metres on the thirdday and so on.(i) Write down a formula for the distance she swims on he nth day.a 750 Find a particular term, Tnd 100 Tn a n 1d 13. The sum of the first n terms of an arithmetic series is; nS n a l (use when the last term l Tn is known) 2 nSn 2a n 1d (use when the difference d is known) 22007 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 sheswims 100 metres more than the previous day.That is she swims 850 metres on the second day, 950 metres on the thirdday 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 1100d 100 Tn a n 1d 14. The sum of the first n terms of an arithmetic series is; nS n a l (use when the last term l Tn is known) 2 nSn 2a n 1d (use when the difference d is known) 22007 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 sheswims 100 metres more than the previous day.That is she swims 850 metres on the second day, 950 metres on the thirdday 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 1100d 100 Tn a n 1d Tn 650 100n 15. (ii) How far does she swim on the 10th day? (1) 16. (ii) How far does she swim on the 10th day? (1) Find a particular term, T10 17. (ii) How far does she swim on the 10th day?(1) Find a particular term, T10 Tn 650 100n 18. (ii) How far does she swim on the 10th day? (1) Find a particular term, T10 Tn 650 100nT10 650 100(10)T10 1650 she swims 1650 metres on the 10th day 19. (ii) How far does she swim on the 10th day?(1) Find a particular term, T10 Tn 650 100nT10 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) 20. (ii) How far does she swim on the 10th day?(1) Find a particular term, T10 Tn 650 100nT10 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, S10n S n 2a n 1d 2 21. (ii) How far does she swim on the 10th day?(1) Find a particular term, T10 Tn 650 100nT10 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, S10n S n 2a n 1d 210S10 2(750) 9(100) 2S10 5 1500 900 12000 22. (ii) How far does she swim on the 10th day? (1) Find a particular term, T10 Tn 650 100nT10 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, S10nn S n 2a n 1d Sn a l 2 OR 210S10 2(750) 9(100) 2S10 5 1500 900 12000 23. (ii) How far does she swim on the 10th day?(1) Find a particular term, T10 Tn 650 100nT10 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, S10n n S n 2a n 1d Sn a l 2 OR21010S10 2(750) 9(100) S10 750 1650 2 2S10 5 1500 900 12000 12000 24. (ii) How far does she swim on the 10th day?(1) Find a particular term, T10 Tn 650 100nT10 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, S10n n S n 2a n 1d Sn a l 2 OR21010S10 2(750) 9(100) S10 750 1650 2 2S10 5 1500 900 12000 12000 she swims a total of 12000 metres in 10 days 25. (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? 26. (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, SnnS n 2a n 1d 2 27. (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, SnnS n 2a n 1d 2 n 34000 1500 n 1100 2 28. (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, SnnS n 2a n 1d 2 n 34000 1500 n 1100 2 68000 n 1400 100n 68000 1400n 100n 2100n 2 1400n 68000 0n 2 14n 680 0 29. (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, SnnS n 2a n 1d 2 n 34000 1500 n 1100 2 68000 n 1400 100n 68000 1400n 100n 2100n 2 1400n 68000 0n 2 14n 680 0 n 34 n 20 0 n 34orn 20 30. (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, SnnS n 2a n 1d 2 n 34000 1500 n 1100 2 68000 n 1400 100n 68000 1400n 100n 2100n 2 1400n 68000 0n 2 14n 680 0 n 34 n 20 0 n 34orn 20 it takes 20 days to swim 34 kilometres 31. Formulae for Geometric Series A series is called a geometric series if; 32. Formulae for Geometric Series A series is called a geometric series if;Tn r Tn1 where r is a constant, called the common ratio 33. Formulae for Geometric Series A series is called a geometric series if;Tn r Tn1 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 n1 34. Formulae for Geometric Series A series is called a geometric series if;Tn r Tn1 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 n1 Three numbers a, x and b are terms in a geometric series if; b x i.e. x ab x a 35. Formulae for Geometric Series A series is called a geometric series if;Tn r Tn1 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 n1 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 36. Formulae for Geometric Series A series is called a geometric series if;Tn r Tn1 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 n1 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 1a 1 r n Sn (easier when r 1)1 r 37. e.g. A companys sales are declining by 6% every year, with 50000 items sold in 2001.During which year will sales first fall below 20000? 38. e.g. A companys sales are declining by 6% every year, with 50000 items sold in 2001.During which year will sales first fall below 20000?a 50000 39. e.g. A companys sales are declining by 6% every year, with 50000 items sold in 2001.During which year will sales first fall below 20000?a 50000r 0.94 40. e.g. A companys sales are declining by 6% every year, with 50000 items sold in 2001.During which year will sales first fall below 20000?a 50000r 0.94 Tn 20000 41. e.g. A companys 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 n1 20000 50000 0.94 n1r 0.94 Tn 20000 42. e.g. A companys 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 n1 20000 50000 0.94 n1r 0.940.4 0.94 n1 Tn 20000 0.94 0.4 n1 43. e.g. A companys 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 n1 20000 50000 0.94 n1r 0.940.4 0.94 n1 Tn 20000 0.94 0.4n1log 0.94 log 0.4n1 44. e.g. A companys sales are declining by 6% every year, with 50000 items sold in 2001.During which year will sales first fall below 20000?a 50000Tn ar n120000 50000 0.94 n1r 0.940.4 0.94 n1 Tn 20000 0.94 0.4 n1 log 0.94 log 0.4 n1 n 1 log 0.94 log 0.4 45. e.g. A companys sales are declining by 6% every year, with 50000 items sold in 2001.During which year will sales first fall below 20000?a 50000Tn ar n120000 50000 0.94 n1r 0.940.4 0.94 n1 Tn 20000 0.94 0.4 n1 log 0.94 log 0.4n1 n 1 log 0.94 log 0.4log 0.4 n 1 log 0.94 n 1 14.80864248n 15.80864248 46. e.g. A companys 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 n120000 50000 0.94 n1 r 0.940.4 0.94 n1Tn 20000 0.94 0.4 n1 log 0.94 log 0.4n1 n 1 log 0.94 log 0.4log 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 47. 2005 HSC Question 7a)Anne and Kay are employed by an accounting firm.Anne accepts employment with an initial annual salary of $50000. Ineach of the following years her annual salary is increased by $2500.Kay accepts employment with an initial annual salary of $50000. in eachof the following years her annual salary is increased by 4% 48. 2005 HSC Question 7a)Anne and Kay are employed by an accounting firm.Anne accepts employment with an initial annual salary of $50000. Ineach of the following years her annual salary is increased by $2500.Kay accepts employment with an initial annual salary of $50000. in eachof the following years her annual salary is increased by 4%arithmetic a = 50000 d = 2500 49. 2005 HSC Question 7a)Anne and Kay are employed by an accounting firm.Anne accepts employment with an initial annual salary of $50000. Ineach of the following years her annual salary is increased by $2500.Kay accepts employment with an initial annual salary of $50000. in eachof the following years her annual salary is increased by 4%arithmetic geometric a = 50000 d = 2500 a = 50000 r = 1.04 50. 2005 HSC Question 7a)Anne and Kay are employed by an accounting firm.Anne accepts employment with an initial annual salary of $50000. Ineach 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 geometrica = 50000 d = 2500a = 50000r = 1.04(i) What is Annes salary in her thirteenth year? (2) 51. 2005 HSC Question 7a)Anne and Kay are employed by an accounting firm.Anne accepts employment with an initial annual salary of $50000. Ineach of the following years her annual s...