The document discusses arithmetic series and provides formulae for calculating terms and sums of arithmetic series. It gives the definition of an arithmetic series as one where the difference between consecutive terms is constant. Formulae are provided for the nth term of an arithmetic series and the sum of the first n terms. An example problem is then given involving calculating the distance swum on the 10th day and the total distance swum in the first 10 days by someone swimming an arithmetic series of distances.

1. Applications of Arithmetic &
Geometric Series
Formulae for Arithmetic Series

2. 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

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
The general term of an arithmetic series with first term a and common
difference d is;
Tn  a  n  1d

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;
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

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)
2
n
Sn  2a  n  1d  (use when the difference d is known)
2

7. 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.

8. 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.

9. 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

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
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

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
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

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.
(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

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.
a  750 Find a particular term, Tn Tn  750   n  1100
d  100 Tn  a  n  1d

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 Find a particular term, Tn Tn  750   n  1100
d  100 Tn  a  n  1d 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  100n
T10  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  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)

20. (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

21. (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

22. (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

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
 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

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
(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

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, Sn
n
S n  2a  n  1d 
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, Sn
n
S n  2a  n  1d 
2
n
34000  1500   n  1100
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, 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

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, 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

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, 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

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
Tn1
where r is a constant, called the common ratio

33. 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

34. 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

35. 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

36. 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

37. 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?

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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

46. 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

47. 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%

48. 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

49. 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

50. 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)

51. 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

52. 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

53. (ii) What is Kay’s annual salary in her thirteenth year? (2)

54. (ii) What is Kay’s annual salary in her thirteenth year? (2)
Find a particular term, T13
Tn  ar n1

55. (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

56. (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?

57. (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

58. (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

59. (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 Sn 
2 n 1
20
S20  2(50000)  19(2500)
2
S10  10 100000  47500 
 1475000

60. (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 Sn 
2 n 1
20
S20  2(50000)  19(2500) 50000 1.0420  1
2 S 20 
0.04
S10  10 100000  47500   1488903.93
 1475000

61. (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 Sn 
2 n 1
20
S20  2(50000)  19(2500) 50000 1.0420  1
2 S 20 
0.04
S10  10 100000  47500   1488903.93
 1475000  Kay is paid $13903.93 more than Anne

62. The limiting sum S exists if and only if  1  r  1, in which case;
a
S 
1 r

63. 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

64. 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

65. 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

66. 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

67. 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

68. 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

69. 2 2 1
S  
1 2  1 1

70. 2 2 1
S  
1 2  1 1
2 2 1
 
1 2
 2  
2 1
 2 2

71. 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

72. 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

73. 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

74. 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

75. 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

76. 2004 HSC Question 9a)
Consider the series 1  tan 2   tan 4   

77. 2004 HSC Question 9a)
Consider the series 1  tan 2   tan 4   
a 1 r   tan 2 

78. 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)

79. 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 

80. 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 

81.  
(ii) For what values of  in the interval    (2)
2 2
does the limiting sum of the series exist?

82.  
(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

83.  
(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

84.  
(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

85.  
(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

86.  
(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

87.  
(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

88.  
(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

89.  
(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