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 Tn1
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 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;
5.
Applications of Arithmetic &
Geometric Series
Formulae 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
6.
Applications of Arithmetic &
Geometric Series
Formulae 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;
7.
Applications of Arithmetic &
Geometric Series
Formulae 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;
bx xa
8.
Applications of Arithmetic &
Geometric Series
Formulae 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;
ab
bx xa 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 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)
2
n
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)
2
n
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)
2
n
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)
2
n
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)
2
n
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)
2
n
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)
2
n
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)
2
n
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, 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 1d
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 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, 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 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, 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 1d 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 1d 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 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
2
n
34000 1500 n 1100
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 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
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
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
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
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
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
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
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
Tn1
where r is a constant, called the common ratio
43.
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;
44.
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
45.
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
x a
46.
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
47.
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;
48.
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
49.
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
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 n1
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 n1
20000 50000 0.94
n1
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 n1
20000 50000 0.94
n1
r 0.94
0.4 0.94
n1
Tn 20000
0.94 0.4
n1
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 n1
20000 50000 0.94
n1
r 0.94
0.4 0.94
n1
Tn 20000
0.94 0.4
n1
log 0.94 log 0.4
n1
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 n1
20000 50000 0.94
n1
r 0.94
0.4 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
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 n1
20000 50000 0.94
n1
r 0.94
0.4 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
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 n1
20000 50000 0.94
n1
r 0.94
0.4 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
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 n1
20000 50000 0.94
n1
r 0.94
0.4 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
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 1d
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 1d
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 1d
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 n1
74.
(ii) What is Kay’s annual salary in her thirteenth year? (2)
Find a particular term, T13
Tn ar n1
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 n1
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 n1
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 n1
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 n1
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 1d
2
79.
(ii) What is Kay’s annual salary in her thirteenth year? (2)
Find a particular term, T13
Tn ar n1
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 1d
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 n1
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 1d 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 n1
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 1d 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 n1
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 1d 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
a2 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
a2 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
a2 1
2 1
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
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