1

Unit 6 Sequences & Series

General Outcome:

• Develop algebraic and graphical reasoning through the study of relations.

Specific Outcomes:

6.1 Analyze arithmetic sequences and series to solve problems.

6.2 Analyze geometric sequences and series to solve problems.

Topics

• Arithmetic Sequences (Outcome 6.1) Page 2

• Arithmetic Series (Outcome 6.1) Page 12

• Geometric Sequences (Outcome 6.2) Page 21

• Geometric Series (Outcome 6.2) Page 27

• Infinite Geometric Series (Outcome 6.2) Page 35 Their Graphs

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Unit 6 Sequences & Series

Sequences:

A sequence is an ordered list of numbers that follow a specific

pattern. The sequence may stop or may continue on indefinitely.

Ex) 5, 7, 9, 11, 13, …

7, 21, 63, 567, …

100, 50, 25, 12.5, …

Notation:

The terms within a sequence are named in order using the

notation 1t , 2t , 3t , etc.

Ex) 1, 4, 9, 16, 25, 36, …

1t , 2t , 3t , 4t , 5t , 6t

Ex) 3, 7, 11, 15, 19, 23, 27, 31, 35, …

4t = 7t =

3

Representing Sequences:

To represent a sequence we can simply list out the first few

terms

Ex) 4, 20, 100, 500, …

or we can use a formula that describes how each term is

generated.

General Term:

The general term is a formula used to describe a sequence that is

based on the term number. (If you want to know the value of the

5th term you put a 5 into the formula.)

Ex) 3 2nt n= − , find the first 3 terms

2 1nt n= + , find the 23

rd term

7 5nt n= − , which term is equal to 51?

4

Types of Sequences:

There are tow main types of sequences we will consider in this

course, they are:

Arithmetic Sequences Geometric Sequences

Is a sequence that Is a sequence that

increases (or decreases) increases (or decreases)

by a common difference. by a common ratio.

Ex) 7, 13, 19, 25, … Ex) 2, 6, 18, 54, …

Arithmetic Sequences:

Is a sequence in which the difference between consecutive terms

is a constant. This constant is called the common difference.

Ex) 3, 5, 7, 9, 11, 13, …

5− , 1− , 3, 7, 11, …

2, 9, 16, 23, 30, …

The general term of an arithmetic sequence can be found using

the formula:

a = value of the first term

d = common difference between terms

( 1)nt a n d= + −

5

Ex) Find the general term of the following sequences.

a) 8, 12, 16, … b) 3, 8, 13, 18, … c) 23, 19, 15, 11, …

Ex) Determine the indicated term for each of the following.

a) 8, 11, 14, 17, … find 31t

b) 37, 30, 23, 16, … find 74t

Ex) Determine how many terms are indicated by each of the

following sequences.

a) 20− , 9− , 2, 13, … …, 442

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b) 31, 36, 41, … …, 791

Arithmetic Means:

Arithmetic means are the terms between two non-consecutive

terms in an arithmetic sequence.

Ex) 2, 5, 8, 11, 14, 17, 20, 23

The arithmetic means between 2 and 23 are

5, 8, 11, 14, 17, 20

The arithmetic means between 5 and 17 are

Ex) Find the four arithmetic means between 27 and 47.

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Ex) Determine the general term for the arithmetic sequence

that has 5 53t = and 17 125t = as two of its terms.

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Arithmetic Sequences Assignment:

1) In each case given below, state whether or not the sequence is arithmetic. If the

sequence is in fact arithmetic, state the values of 1t and d.

a) 16, 32, 48, 64, 80, … b) 2, 4, 8, 16, 32, …

c) 4, 7, 10, 13, 16, ...− − − − − d) 3, 0, 3, 6, 9, ...− − −

2) Write the first four terms of the following arithmetic sequences given 1t and d.

a) 1 5t = , 3d = b) 1 1t = − , 4d = −

c) 1 4t = , 1

5d = d) 1 1.25t = , 0.25d = −

3) Determine the value of the indicated term for each sequence below.

a) 3 8nt n= + , determine 9t b) 4 22nt n= − + , determine 6t

9

c) 2 4

5 5nt n= + , determine 12t d)

3 3

2 5nt n

−= − , determine 27t

4) Determine the number of terms represented by each of the following arithmetic

sequences.

a) 4, 2, 8, ... ..., 170− b) 1 4

2 , 2, 1 , ... ..., 145 5

−

c) 3, 1, 5, ... ..., 97− d) 14, 12.5, 11, ... ..., 10−

5) Determine the general term for the following arithmetic sequences.

a) 5, 9, 13, 17, ... b) 11, 4, 3, 10, ...− −

6) Determine the two arithmetic means between 6 and 33.

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7) Determine the four arithmetic means between 8 and 63.

8) Determine the three arithmetic means between 42 and 18−

9) Determine with arithmetic sequences given below contain the term 34.

a) 6 ( 1)4nt n= + − b) 3 1nt n= −

c) 12a = , 5.5d = d) 3, 7, 11, …

10) Determine the value of the first term of the arithmetic sequence in which

16 110t = and the common difference is 7.

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11) The first three terms of an arithmetic sequence are given below. Determine the

value of x and state the values of the first three terms.

5 2, 7 4, 10 6x x x+ − +

12) The numbers x, y, and z are the first three terms of an arithmetic sequence.

Express z in terms of x and y.

13) If 9 73t = and 15 145t = are terms of an arithmetic sequence, determine the

general term for the sequence.

14) Susan joined a fitness class at her local gym. Into her workout, she

incorporated a sit-up routine that followed an arithmetic sequence. On the 6th

day of the program, Susan completed 11 sit-ups. On the 15th day she completed

29 sit-ups. Determine the general term that relates the number of sit-ups to the

number of days. On which day will Susan be able to do 100 sit-ups?

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Arithmetic Series:

When the terms of sequence are added together we call it a

series

Ex) 3, 5, 7, 9, 11, 13, … is an arithmetic sequence

3 5 7 9 11 13 ...+ + + + + + is an arithmetic series

Notation:

The partial sums of a series are denoted using the following

notation:

5S denotes the sum of the first 5 terms

31S denotes the sum of the first 31 terms

Ex) Given the series below, find the following partial sums.

3 5 7 9 11 13 ...+ + + + + +

a) 3S b) 7S c) 79S

Ex) What is the sum of the first 100 natural numbers?

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From this thinking we can develop a formula to determine the

sum of an arithmetic series.

But ( 1)nt a n d= + − so

( 1)2

n

nS a a n d= + + −

Ex) Find the sum of the first 70 terms of the series given

below.

8 15 22 ...+ + +

Ex) Determine 37S for the series given by 4 10 16 ...+ + +

( )2

n n

nS a t= +

2 ( 1)2

n

nS a n d= + −

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Ex) Determine the sum of each series given below.

a) 17 30 43 56 ... ... 563+ + + + +

b) 249 243 237 ... ... 27+ + + +

Ex) Determine 11S for the arithmetic series that has 1 9t = − and

11 31t = as two of its terms.

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Ex) If 1 25t = , 382nt = , and 4477nS = for a particular series,

determine the number of terms n represented in the partial

sum.

Ex) The sum of the first three terms of an arithmetic series is

36 and the sum of the first four terms is 76. Determine the

first six terms of the series and 6S

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Arithmetic Series Assignment:

1) Determine the sum of each arithmetic series.

a) 5 8 11 ... ... 53+ + + + b) 7 14 21 ... ... 98+ + + +

c) ( ) ( )8 3 2 ... ... 102+ + − + − d) 2 5 8 41

... ...3 3 3 3+ + + +

e) 1 3 5 ...+ + + , 8S f) 40 35 30 ...+ + + , 11S

g) 1 3 5

...2 2 2+ + + , 7S h) ( ) ( )3.5 1.25 1− + − + +…, 6S

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2) Determine the sum for each arithmetic sequence described below.

a) 1 7t = , 79nt = , 9n = b) 58a = , 7nt = − , 26n =

c) 12a = , 8d = , 9n = d) 1 42t = , 5d = − , 14n =

3) Determine the value of the first term, a, for each arithmetic series described

below.

a) 6d = , 574nS = , 14n = b) 6d = − , 32nS = , 13n =

c) 0.5d = , 218.5nS = , 23n = d) 3d = − , 279nS = , 18n =

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4) Determine the number of terms, n, that are represented in each arithmetic series

below.

a) 8a = , 68nt = , 608nS = b) 1 6t = − , 21nt = , 75nS =

5) Determine the sum of all the multiples of 4 between 1 and 999.

6) If 2 40t = and 5 121t = , determine the sum of the first 25 terms of the series.

7) If 2 40S = and 5 121S = , determine the sum of the first 25 terms of the series.

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8) The sum of the first five terms of an arithmetic series is 85. The sum of the first

six terms is 123. Determine the values of the first four terms in the series.

9) The first three terms of an arithmetic sequence are given below.

, 2 5, 8.6x x −

a) Determine the value of x.

b) Determine the values of 1t and d.

c) Determine the value of 20S

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10) A number of interlocking rings each 1 cm thick are hanging from a peg. The

top ring has an outside diameter of 20 cm. The outside diameter of each of the

outer rings is 1 cm less than that of the ring above it. The bottom ring has an

outside diameter of 3 cm. What is the distance form the top of the top ring to

the bottom of the bottom ring?

11) The 15th term in an arithmetic sequence is 43 and the sum of the first 15 terms

of the series is 120. Determine the first three terms of the series.

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Geometric Sequences:

Geometric sequences are sequences where there is a common

ratio between successive terms.

Ex) 1, 3, 9, 27, 91, … 3 2 , 6, 6 2 , …

Common ratio = Common ration =

The general term for a geometric sequence can be found with

the following formula:

a = value of first term

r = common ratio

Ex) Determine the general term for the sequence given by

5, 10, 20, 40, 80, … Find the value of 8t .

1n

nt ar−=

22

Ex) How many terms are represented by the sequence

4, 20, 100, … …, 39 062 500

Ex) A car is worth $30 000 when first purchased. Each year the

car depreciates 15%. How much will it be worth in 8

years?

Ex) Determine the values of the 3 geometric means between

147 and 352947.

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Ex) Determine the general term for the geometric sequence

where 3 36t = and 10 78732t = .

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Geometric Sequences Assignment:

1) Determine whether or not the following are examples of geometric sequences.

If it is, state determine the common ratio and the general term for the sequence.

a) 1, 2, 4, 8, … b) 2, 4, 6, 8, …

c) 3, 9, 27, 81, ...− − d) 1, 1, 2, 4, 8, …

e) 10, 15, 22.5, 33.75, … f) 1, 5, 25, 125, ...− − − −

2) Determine the first four terms for each geometric sequence.

a) 1 2t = , 3r = b) 1 3t = − , 4r = −

c) ( )1

4 3n

nt−

= − d)

11

22

n

nt

−

=

25

3) Determine the 4 geometric means between 3 and 3072

4) Determine the 3 geometric means between 8.1 and 240.1.

5) Determine the general term for each of the geometric sequences given below.

a) 2r = , 3a = b) 192, 48, 12, 3, ...− −

c) 3 5t = , 6 135t = d) 1 4t = , 13 16384t =

6) Determine the value of y for the geometric sequence given below.

3, 12, 48, 5 7y +

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7) The colour of some clothing fades over time when washed. Suppose a pair of

jeans fades by 5% with each washing.

a) What percent of the colour remains after one washing?

b) If 1 100t = , what are the first four terms of the sequence?

c) What is the value of r for this geometric sequence?

d) What percent of the colour remains after 10 washings?

e) How many washings would it take so that there is only 25% of the original

colour remaining in the jeans?

8) Given the geometric sequence below, determine the value of x, and determine

the value of the first three terms of the sequence.

2, 2 1, 4 3x x x+ + −

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Geometric Series:

A geometric series is the sum of the terms of a geometric

sequence.

Ex) 2, 8, 32, 128, … is a geometric sequence

2 8 32 128 ...+ + + + is a geometric series

The Partial Sum of a geometric series can be found using :

or

nS = sum of terms

a = value of the first term

r = common ratio

n = number of terms considered

nt = value of last term considered

Ex) Find the sum of the first 12 terms for the sequence given

by 1 4 16 64 ...+ + + +

( )11

n

n

a rS

r

−=

− 1

nn

rt aS

r

−=

−

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Ex) Determine the sum of the first 8 terms of the geometric

series given below.

5 15 45 ...+ + +

Ex) Find the sum of each geometric series given below.

a) 3 6 12 24 ... ... 49512 98304− + − + + −

b) 1 1 1 1

... ...9 27 81 19683+ + + +

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Ex) In a Scrabble tournament there are 256 entries. In round

one of the tournament each of the entries are paired up to

play. The winner of each match then goes on to the second

round of play with the losers being eliminated. This

process is then repeated in each successive round until the

final match. For this tournament, how many total matches

will be played and how many rounds will the tournament

take to complete?

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Geometric Series Assignment:

1) Determine whether each series is geometric. Justify your answer.

a) 4 24 144 864 ...+ + + + b) 40 20 10 5 ...− + − + −

c) 3 9 18 54 ...+ + + + d) 10 11 12.1 13.31 ...+ + + +

2) Determine the sum for each geometric series given below. Express your

answers as exact values in fraction form.

a) 6 9 13.5 ...+ + + determine 10S b) 18 9 4.5 ...− + − determine 12S

c) 2.1 4.2 8.4 ...+ + + determine 9S d) 0.5 0.05 0.005 ...+ + +

determine 5S

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e) 1 12t = , 2r = , 6144nt = f) 1 27t = , 1

3r = ,

1

81nt =

g) 1

256a = , 4r = − , 1024nt = − h) 72a = ,

1

2r = ,

9

256nt =

3) Determine the sum, nS , for each geometric series given below. Round your

answers to the nearest hundredth if necessary.

a) 1

27 9 3 ... ...243

+ + + + b) 1 2 4 128

... ...3 9 27 6561+ + + +

c) 1 5t = , 81920nt = , 4r = d) 3a = , 46875nt = , 5r = −

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4) Determine the value of 1t for each geometric series described below.

a) 33nS = , 48nt = , 2r = − b) 6 443S = , 1

3r =

5) The sum of 4 12 36 108 ... ... nt+ + + + + is 4372. Determine how many terms

are in the series.

6) The common ratio of a geometric series is 1

3 and the sum of the first 5 terms is

121. Determine the value of the first term.

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7) A fan out system is used to contact a large group of people. The person in

charge of the contact committee relays the information to four people. Each of

these four people notifies four more people, who in turn each notify four more

people, and so on.

a) Write the corresponding series for the number of people contacted.

b) Determine how many people are notified after 10 levels of this system.

8) Celia is training to run a marathon. In the first week she runs 25 km and

increases this distance by 10% each week. This situation may be modeled by

the series 225 25(1.1) 25(1.1) ...+ + + She wishes to continue this pattern for 15

weeks. Determine how far will she have run in total when she completes the

15th week. Round your answer to the nearest tenth of a km.

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9) Determine the number of terms that are represented by the following geometric

series.

2 33 3 3 ... ... 3 9840n+ + + + =

10) The third term of a geometric series is 24 and the forth term is 36. Determine

the sum of the first 10 terms. Express your answer as a reduced fraction.

11) If 7 89S = and 8 104S = , determine the value of 8t .

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Infinite Geometric Series:

The sum of an infinite list of numbers (an infinite series) will do

one of two things:

• Converge → Approach a specific value

• Diverge → Approach positive or negative infinity or bounce between 2 or

more values

Ex) State whether the following series are convergent or

divergent.

a) 2 2 2 2 2 2 2 ...− + − + − + − +

b) 1 1

8 4 2 1 ...2 4

+ + + + + +

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An infinite geometric series will converge if 1 1r− .

Proof: ( )1

1

n

n

a rS

r

−=

− sum of a geometric

Series

Ex) If possible, find the sum of each series given below.

a) 1 1

1 ...4 16

+ + + b) 1

9 3 1 ...3

+ + + +

c) 5 20 80

...3 9 27+ + + d)

3 33 ...

5 25− + −

37

Ex) Express 0.35 as a fraction.

Ex) Express 2.421 as a fraction.

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Infinite Geometric Series Assignment:

1) Determine whether each infinite geometric series is convergent or divergent.

a) 1 3t = − , 4r = b) 1 4t = , 1

4r

−=

c) 125 25 5 ...+ + + d) 243 81 27 9

...3125 625 25 5

− + − +

2) Determine the sum of each infinite series if possible.

a) 1 8t = , 1

4r

−= b) 1 3t = ,

4

3r =

c) 12 36 108

4 ...5 25 125

− + − + d) 1280 640 320 160

...729 243 81 27

+ + + +

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e)

2 32 2 2

5 5 5 5 ...3 3 3

+ + + +

f)

2 31 1 1

1 ...4 4 4

− − − + + + +

3) If the sum of an infinite geometric series is 81 and the its common ratio is 2

3,

determine the value of the first term.

4) The first term of an infinite geometric series is 8− and its sum is 40

3

−.

Determine the value of the common ratio.

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5) The sum of the infinite geometric series shown below is 4. Determine the value

of x.

2 31 3 9 27 ...x x x+ + + +

6) The sum of an infinite geometric series is twice its first term. Determine the

value of the common ratio.

7) Each of the following represents an infinite geometric series. For what values

of x will each series be convergent?

a) 2 35 5 5 5 ...x x x+ + + + b)

2 3

1 ...3 9 27

x x x+ + + +

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Answers

Arithmetic Sequences Assignment:

1. a) arithmetic, 1 16t = , 16d = b) not arithmetic

c) arithmetic, 1 4t = − , 3d = − d) arithmetic, 1 3t = , 3d = −

2. a) 5, 8, 11, 14 b) 1, 5, 9, 13− − − − c) 21 22 23

4, , , 5 5 5

d) 1.25, 1, 0.75, 0.5

3. a) 9 35t = b) 6 2t = − c) 1228

5t = d) 27

411

10t

−=

4. a) 30n = b) 82n = c) 26n = d) 17n =

5. a) 4 1nt n= + b) 7 18nt n= − +

6. 15, 24

7. 19, 30, 41, 52

8. 27, 12, 3−

9. a) yes b) no c) yes d) no

10. 1 5t =

11. 16x = − , 78, 116, 154− − −

12. 2z y x= −

13. 12 35nt n= −

14. 2 1nt n= − , 51 days

Arithmetic Series Assignment:

1. a) 17 493S = , b) 14 735S = c) 23 1081S = − d) 14301

3S =

e) 8 64S = f) 11 165S = g) 749

2S = h) 6 12.75S =

2. a) 387nS = b) 663nS = c) 396nS = d) 133nS =

3. a) 2a = b) 500

13a = c) 4a = d) 41a =

4. a) 16n = b) 10n =

42

5. 124500

6. 25 8425S =

7. 25 1305S =

8. 3 10 17 24+ + +

9. a) 6.2x = b) 1 6.2t = , 1.2d = c) 20 352S =

10. 173 cm

11. ( ) ( ) ( )27 22 17− + − + −

Geometric Sequences Assignment:

1. a) geometric, 2r = , 12nnt−= b) not geometric

c) geometric, 3r = − , ( )1

3 3n

nt−

= − d) not geometric

e) geometric, 3

2r = ,

13

102

n

nt

−

=

f) geometric, 5r = , ( )

15

n

nt−

= −

2. a) 2, 6, 18, 54 b) 3, 12, 48, 192− − c) 4, 12, 36, 108− −

d) 1 1

2, 1, , 2 4

3. 12, 48, 192, 768

4. 189 441 1029

, , 10 10 10

or 189 441 1029

, , 10 10 10

− −

5. a) ( )1

3 2n

nt−

= b)

11

1924

n

nt

−−

=

c) ( )15

39

n

nt−

=

d) ( )1

4 2n

nt−

=

6. 37y =

7. a) 95% b) 95, 90.25, 85.7375, 81.450625 c) 0.95 d) 59.87%

e) approximately 28 washings

8. 7x = , 9, 15, 25

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Geometric Series Assignment:

1. a) yes, 6r = b) yes, 1

2r

−= c) no d) yes, 1.1r =

2. a) 10174075

256S = b) 12

12285

1024S = c) 9

10731

10S = d) 5

11111

20000S =

e) 12276nS = f) 3280

81nS = g)

209715

256nS

−= h)

36855

256nS =

3. a) 9841

40.50243

nS = b) 6305

0.966561

nS = c) 109225nS =

d) 39063nS =

4. a) 1 3t = b) 1107649

364t =

5. 7n =

6. 1 81t =

7. a) 4 16 64 256 ...+ + + + b) 1398100 people

8. 794.3 km

9. 8n =

10. 1058025

48S =

11. 4 15t =

Infinite Geometric Series Assignment:

1. a) Diverges b) Converges c) Converges d) Diverges

2. a) 32

5S = b) Diverges c)

5

2S = d) Diverges e) 15S =

f) 4

5S =

3. 1 27t = 4. 2

5r = 5.

1

4x = 6.

1

2r =

7. a) 1 1x− b) 3 3x−