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Sequence & Series - Arithmetic

Sequences & Series

Arithmetic


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S1SERIES TOPIC NUMBER

L 1

Sequences and Series: Arithmetic

What is the dierence between a sequence and a series?

What is an arithmec sequence?

What is the formula for the general term of an arithmec sequences?

Suggest one praccal applicaon for the sum of an arithmec series.

What is the dierence between a sequence and a series?

What is an arithmec sequence?

What is the formula for the general term of an arithmec sequences?

Suggest one praccal applicaon for the sum of an arithmec series.

Answer these quesons, beforeworking through the chapter.

Answer these quesons, afterworking through the chapter.

But now I think:

What do I know now that I didnt know before?

I used to think:

Sequences and series of numbers occur frequently in real life and mathemacs. This booklet introduces thebasic concepts focusing on arithmec sequences and series. Several applicaons will also be discussed

Sequences and Series: Arithmetic


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L2

Sequences and Series: Arithmetic Basics

A sequenceis a list of numbers in a specic order like this

3, 5, 8, 9

Each number in a sequence is called a termof the sequence.

Above, the 1st term is 3and the 2nd term is 5. We write this as T 31 = , T 52 = and so on.

Here are two examples.

Look at these sequences.

a

a b

b4, 7, 34, 0, 9, 7 -10, 14, 20, -22, 26

`The rst 5terms are 9, 13, 17, 21, 25. `The rst 5terms are 4, -1, -6, -11, -16.

10T1 =- rst term

14T2 = second term

T 203 = third term

T 224 =- fourth term

The number of terms of the sequence is 5.

T 41 = rst term

T 72 = second term

T 343 = third term

T 04 = fourth term

The number of terms of the sequence is 6.

T1 is the 1st term of the sequence, T2 is the 2

nd term of the sequence. Tn is the nth term of the sequence, also called

the General term of the sequence.

Here are two examples.

Write down the rst 5terms of these sequence with the formulas given.

The nth term of a sequence is given by the

formula T n4 5n = + . Use this with , , , ,1 2 3 4 5n = to write down the rst 5terms of this sequence.

The nth term of a sequence is given by the formula

T n9 5n = - . Use this with , , , ,1 2 3 4 5n = to writedown the rst 5terms of this sequence.

4 5

4 5

4 5

4 5

4 5

T

T

T

T

T

1 9

2 13

3 17

4 21

5 25

1

2

3

4

5

= + =

= + =

= + =

= + =

= + =

^

^

^

^

^

h

h

h

h

h

9 5

9 5

9 5

9 5

T

T

T

T

T

9 5 1 4

2 1

3 6

4 11

5 16

1

2

3

4

5

= - =

= - = -

= - = -

= - = -

= - = -

^

^

^

^

^

h

h

h

h

h

This sequence has 4numbers.

The numberof terms of the sequence is 4.

Sequences and Terms

General Term of a Sequence


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L 3


a b

12 5 7

...

T

T

T

T

5

19 5 14

26 5 21

5 7

5 7

5 7

5 7

0

1

2

3

1

2

3

4

#

#

#

#

`

=

= = +

= = +

= = +

= +

= +

= +

= +

...

T

T

T

T

125

115 125 10

105 125 20

95 125 30

125 10

125 10

125 10

125 10

0

1

2

3

1

2

3

4

#

#

#

#

`

=

= = -

= = -

= = -

= -

= -

= -

= -

T

T n

T n

n5 7

5 7 7

7 2

1n

n

n

#

`

= +

= + -

= -

-^ h T

T n

T n

n125 10

125 10 10

135 10

1n

n

n

#

`

= -

= - +

= -

-^ h

Here are two examples where you need to nd the formula for the general term of a sequence.

Find the formula for the general term Tn for the sequences.

The nth term of a sequence is given by T n9 8n = + . For which value of n does the n

th term equal 161?

5, 12, 19, 26, 33, 40, ... 125, 115, 105, 95, 85, ...

The terms are increasing by 7. The terms are decreasing by 10.

Here is an example where a term is given and the nvalue found.

Substung into the formula gives

n

n

n

n

n

161 9 8

9 161 8

9 153

9

153

17

`

= +

= -

=

=

=


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L4

Sequences and Series: Arithmetic Questions Basics

For the sequences shown, ll in the values indicated1.

a

a

c

a

a

a

b

b

b

b

d

b3, 6, 1, -7, 0, 45 9, 6, 2, -3, -10, 20, 4

The number of terms in the sequence = The number of terms in the sequence =

T

T

T

T

1

2

3

4

=

=

=

=

T

T

T

T

1

2

3

4

=

=

=

=

Write down the rst 5terms of the sequence whose general term is given:2.

3.

5.

4.

T n7n =

T n1 3n = -

T n100 8n = + 2 6T n5n = - -

T n6 7n = +

19T nn =-

What is the 5th term of the sequence with nth term:

The nth term of a sequence is given by

T n8 5n = - . For which value of n does

the nth term equal 163?

The nth term of a sequence is given by

T n3 12n = - . For which value of n does

the nth term equal -177?

Try and nd the general term, Tn , of these sequence

2, 4, 6, 8, ... 4, 7, 10, 13, ...


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L 5


T

T

T

T

10

10 5 15

15 5 20

20 5 25

1

2

3

4

=

= + =

= + =

= + =

T T d

2 6

8

2 1` = +

= +

=

T T d

20 6

26

5 4` = +

= +

=

An Arithmec Sequence is a specic type of sequence where the dierence (d) between consecuve terms is

constant (i.e. the same)

Examples of arithmec sequences

A sequence has 10T1 = and a common dierence of 5. Find T2 , T3 , T4 .

If the following sequence is arithmec nd the missing terms: 2, T2 , 14, 20, T5 , 32, 38

a bAn arithmec sequence with 5terms An arithmec sequence with 6terms

4, 8, 12, 16, 20 32, 27, 22, 17, 12, 7

,4T1 =

,8T2 = and so on.

,T 321 =

,27T2 = and so on.

or

.

d

d

T T

T T

8 4

12 8

4

4

2 1

3 2

= -

= -

= -

= -

=

=

or5d

d

T T

T T

27 32

22 27 5

2 1

3 2

= -

= -

= -

= -

= -

= -

The common dierence is 4. The common dierence is -5.

+4 -5+4

d d d

-5+4 -5+4 -5 -5

To find the common difference, find the difference between consecutive terms.

Here is an example where the rst term and common dierence are used to nd other terms.

Start with 10and increase by 5each term.

So the arithmec sequence is: 10, 15, 20, 25.

The common dierence can be used to nd missing terms.

First, nd d, by subtracng tow consecuve terms. d 20 14 6= - =

Use T1 and dto nd T2

Use T4 and dto nd T5

... ...d T T T T T T n n2 1 3 2 1= - = - = = - =-

..., , , ,T T T T 1 2 3 4`

The dierence, d, is called the common dierence, and in general we say: d T Tn n 1= - - . An arithmec sequence

can also be called an Arithmec Progression and the abbreviaon AP is commonly used.

Arithmetic Sequences

Always add or subtract the same number.


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L6


For each of these arithmec sequences below, nd6.

7.

8.

9.

10.

11.

(i) (ii)

(iii) (iv)

the number of terms in the sequence the rst term, T1

the h term, T5 the common dierence, d

a

c

e

a

a

a

b

b

b

b

d

f

12, 16, 20, 24, 28, 32, ...

1, 7, 13, 19, 25

0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30

3, 8, ...

17, ...

5, T2 , 19... -6, T2 , 8...

-19, -14, ...

3, -1, ...

-10, -7, -4, -1, 2, 5, 8

-5, 7, 19, 31, 43, 55

4.2, 5.8, 7.4, 9.0, 10.6, 12.2

What are the next three terms if the common dierence for these sequences is 5?

What are the next three terms if the common dierence for these sequences is -4?

What is the second term if the common dierence for these sequences is 7?

The h and seventh terms of an arithmec sequence are 19and 27respecvely. Find d.

The rst term of an arithmec sequence is 3and d=-4. What is the seventh term of the sequence?


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L 7


The leer a is used to represent the rst term of a sequence. The leer d is used to represent the common

dierence of an arithmec sequence. For example

8, 18, 28, 38, ...

2, 5, 8, 11, 14, 17, ...

The common dierence is d 18 8 10= - =The rst term is a 8=

The next step is to work out a formula for any term of an arithmec sequence. Here is an example.

Finding the general term (also called the nth term) of an arithmec sequence

The sequence 3, 10, 17, 24, 31, 38, 45, 52has a common dierence of d=7

T

T

T

T

3

10 3 7

17 3 7 7 3 2 7

24 3 7 7 7 3 3 7

1

2

3

4

=

= = +

= = + + = +

= = + + + = +

^

^

h

hCan you see the paern?

The general term is given by T n3 7 1n = + -^ h, and this simplies to give T n7 4n = -

Starng from a and adding the common dierence d for each term, we get:

The formula for each term of an arithmec sequence a, a+d, a+2d, a+3d, ... is:

T a1 = T T d

a d

a d2 1

2 1= +

= +

= + -^ h

2 3

T a n d

n

n

T n

1

1

2 3 3

3 1

n

n

#

= + -

= + -

= + -

= -

^

^

h

h

T T d

a d

a d

2

3 1

3 2= +

= +

= + -^ h

T T d

a d

a d

3

4 1

4 3= +

= +

= + -^ h

T T d

a d

a d

4

5 1

5 4= +

= +

= + -^ h

T a n d1n = + -^ h

n=number of the term in the sequencenth term

d=common dierencea=rst term

This is the General Termof an Arithmec Sequence.

Finding the general term of the following arithmec sequence using the formula.

The arithmec sequence

has rst term 2a = , and the common dierence is d 5 2 3= - = , so the formula for the nth term is

The 10th term is given by T 3 10 1 2910 = - =^ h .

The 15th term is given by T 3 15 1 4415 = - =^ h .

General Term of an Arithmetic Sequence


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L8


Here is an example of using the formula to nd d.

The common dierence of an arithmec sequence is 14and the twelh term is 179. Find the rst term.

The common dierence of an arithmec sequence is -8and the twelh term is -144. Find the rst term.

Find dif a 5= and T 717 =

This is given: 5 7 71a n T7= = =

Substung these into the formula T a n d 1n = + -^ h gives:

d

d

d

71 5 7 1

71 5 6

6

71 511

`

`

= + -

= +

= -

=

^ h

`The common dierence is d 11=

Somemes a term that isnt the rst term is used to nd terms of a sequence. In the next example the general

term is needed to nd the rst term, T a1 = .

The next example has a negave d.

This is given:

This is given:

14 179d T12= =

8 16 144d n T16= - = = -



179 14

179 14

a

a

a

12 1

11

25

= + -

= -

=

^

^

h

h

a

a

a

144 16 1 8

144 15 8

6

- = + - -

= - +

=

^ ^

^ ^

h h

h h

So, the rst term is a 25= .

So, the rst term is a 6= .


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L 9


Here is an example of nding the nth term of a sequence.

For the arithmec sequence -12, -7, -2, 3, 8, 13, 18, ...

In a sequence T 1498 = and T 22512 = . Find the general term.

a

c

b

d

Find a

Find an expression for the nth term Tn .

Find d

Find T20

a T 121= = -

12 5

T a n d

n

T n

1

1

17 5

n

n`

= + -

= - + -

= - +

^

^

h

h

d T T 8 3 55 4= - = - =

In the next example, two non-consecuve terms are used to nd the general term of the sequence, Tn.

The general term is used to form a system of two simultaneous equaons which can be solved.

T 17 5 20

83

20 = - +

=

^ h

By substung the informaon into the formula T a n d 1n = + -^ h two equaons are formed, and these needto be solved simultaneously for aand d.

Substung gives us

Solve these equaons:

Subtracng 2 - 1 gives:

Substung into 1 gives:

149 a d8 1` = + -^ h

a d149 7= +

a d225 11= +

225 149 11 7

76 4

d d

d

d 19`

- = -

=

=

225 a d12 1` = + -^ hand

1

2

149 7a

a

19

16`

= -

=

^ h

16 19T n

T n

1

19 3

n

n

`

`

= + -

= -

^ h

Tn=149 , n=8 T12=225 , n=12


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L10


Find the formula for the nth term, Tn , of an arithmec sequence with rst term, a=9, and common

dierence, d=12.

12.

13.

14.

15.

16.

17.

For the arithmec sequence 21, 16, 11, 6, 1, -4, -9:

The h term of an arithmec sequence is T 175 = . Using the formula T a n d1n = + -^ h , and given thatthe rst term is a 1= , nd the common dierence d.

The rst term of an arithmec sequence is 21and the tenth term equals -78. Find d.

A sequence has a common dierence 9and T 17420 = . Find the rst term.

An arithmec sequence has 4T 55 = and 2T 78 = . Find the general term by substung into

T a n d1n = + -^ h and solve a pair of simultaneous equaons for a, d.

a

c

b

d

Find a

Find an expression for the nth term Tn .

Find d

Find T20


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L 11

Sequences and Series: Arithmetic Knowing More

A seriesis the sum of the terms in a sequence.

Comparing a sequence to a series.

Consider the series ...2 4 6 8 10 12 14 16+ + + + + + + +

5, 2, 13, 10, 8is a sequence. 5+2+13+10+8is a series.

Tn is used as a symbol for the nth term. Terms in a series are found the same way as terms in a sequence.

Sn is the notaon for the sum of the rst n terms. Here are some basic examples.

The second term of the sequence is T 42 = .

The sum of the rst two terms of the sequence is wrien

b

a

The h term is T 105 = .

The sum of the rst ve terms is wrien

S T T 2 4 62 1 2= + = + =

S T T T T T 2 4 6 8 10 305 1 2 3 4 5= + + + + = + + + + =

A series has 16terms and is given that S 54015 = and S 60016 = . Find the value of the 16thterm, T16 .

The next example highlights the denion of the sum to nterms.

By denion, ... ...S T T T S T T T T 15 1 2 15 16 1 2 15 16= + + + = + + + +

Subtracng these two leaves us with just T16.

Therefore,

... ...S S T T T T T T T T 16 15 1 2 15 16 1 2 15 16- = + + + + - + + + =^ ^h h

T 600 540 6016 = - =

Series

Given a series with 16terms: 2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53.

Find S1, S4, S10, S16

2S T1 1= =

S T T T T

2 3 5 7

17

4 1 2 3 4= + + +

= + + +

=

To nd S16the sum S10can be used.

2 3 5 7 11 13 17 19 23 29

S T T T T T T T T T T

139

10 1 2 3 4 5 6 7 8 9 10= + + + + + + + + +

= + + + + + + + + +

=

S S T T T T T T

129 31 37 41 43 47 53

381

16 10 11 12 13 14 15 16= + + + + + +

= + + + + + +

=

This example nds the sum to nterms for serveral nvalues.


14/42




L12


Remember that in an arithmec sequence, consecuve terms have the same dierence. This holds in an

arithmec series also, consecuve terms always dier by the same amount, d, the common dierence. The terms

Tn + 1 and Tn are consecuve and for an arithmec series:

Arithmetic Series

Find the missing terms if the given series are arithmec.

Find the missing terms of the series 20 116T T2 3+ + + if it is an arithmec series.

a

a

b

b

T13 30 3+ +

d T T

30 13

17

2 1= -

= -

=

T d30

30 17

47

3 = +

= +

=

9 T 312+ +

and31 9T a3 = =

9 31d

d

3 1

9 2 31

`

`

+ - =

+ =

^ h

Solve for d, d

T

2

31 9 11

9 11 202

= - =

= + =So

Here is an example of an arithmec series with two unknown terms.

Here are some arithmec series.

Consecuve terms have a dierence of d.

and116 20T a4 = =

20 3 116

d

d

d

d

20 4 1 116

3

116 20

32

` + - =

+ =

= -

=

^ h

So the missing terms are

and

T 20 32 522 = + =

T 52 32 843 = + =

Examples of arithmec series.

S 1 9 17 25 33 41 497 = + + + + + + S 10 8 6 43 = + + +

The common dierence is d=9-1=8. The common dierence is d=8-10=-2.

Here are some basic examples to help understand arithmec series

T T d T T d a n d 1n n n n1 1- = = + = + -+ + ^ h


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L 13


Find the sum of the rst 25terms of the arithmec series ...4 10 16 22 28 34+ + + + + +

The third term of an arithmec series is given by T 563 = and the sum of the 3rd

and 4th

terms is T T 1403 4+ = .

a b

What about the general sum, Sn? It is important to nd a formula for Sn based on a, n, and d. Here is the derivaon.

The sum of the rst nterms of an arithmec series with rst term, a, and common dierence, d, is given by

S a

S a n d 1

n

n

=

= + -^^ h h

...

...

...

S S a a n d a d a n d a n d a d a n d d

S a n d a n d a n d a n d

n n

1 2 2 1

2 2 1 2 1 2 1 2 1

1 2 1

n n

n

`

`

+ = + + - + + + + - + + + - + + + + - +

= + - + + - + + + - + + -

-

^ ^ ^ ^ ^ ^

^ ^ ^ ^

h h h h h h

h h h h

6 6 6 6

6 6 6 6

@ @ @ @

@ @ @ @

a d

a n d2

+ +

+ + -

^

^^

h

h h

...

...

a n d

a d

2+ + + -

+ + +

^^

^

h h

h

a n d

a

1+ + -

+

^^ h hIn reverse:

Adding these two rows gives:

S n

a n d2

2 1n = + -^ h6 @

The common dierence d

The rst term a

This is important if you want to nd S100 or S200. Here is an example.

,4a T1= = d 10 4 6= - = n 25=

2

2 6

S n

a n d2

1

2

254 25 1

1900

25 = + -

= + -

=

^

^ ^

h

h h

6

6

@

@

The next example uses the sum of two terms to nd aand d, which are used in the sum formula.

Find the fourth term, T4

56T

T T 140

3

3 4

=

+ =

Subtracng gives T 140 56 844 = - =

Find the sum to 40terms, S40 .

d T T

a T T d T d

84 56 28

2

4 3

1 2 3

= - = - =

= = - = -

a 56 2 28 0` = - =^ h

2

2 28

S n

a n d

S

2 1

2

400 40 1

21840

40

40

= + -

= + -

=

^

^ ^

h

h h

6

6

@

@

The sum to 25terms is 1900.

The number of terms, n

Sum of an Arithmetic Series

There are nlots of a n d2 1+ -^^ h h

S n a n d 2 2 1n` = + -^ h6 @

Finally dividing by 2gives the general term.

Sum to nterms:


16/42




L14


Find the sum of the arithmec series ...5 10 15 20 25 2555+ + + + + + .

An arithmec series has sum S n n6

51n = +^ h. How many terms must be taken for the sum to exceed 200?

The sum of the rst neven numbers is an arithmec series and ...S n2 4 6 8 2n = + + + + +

a 5= 5d= 5l 255=

Step 1: Find n. Step 2: Substute known values

T a n d

n

n

1

5 1 5

5

n = + -

= + -

=

^

^

h

h

T

n

n

2555

5 2555

5

2555511

n

`

=

=

= =

S n

a l2

2

5115 2555

654080

511` = +

= +

=

^

^

h

h

If the last term of the series lis known then the formula 2S n a n d2

1n = + -^ h6 @becomes

S n

a l2

n = +^ h

The last term is l a n d 1= + -^ hThe rst term a

Somemes nneeds to be found before being able to calculate Sn. Use the formula for the last term Tnto nd thenumber of terms, n.

a bWhat is the general term,Tn? Use the formula S n

a l

2n = +

^ hto nd the

formula for the sum to nterms,Sn.The last term in the series sum gives away

the general term, T n2n` = .

Check that the formula gives 2, 4, 6, 8

when substung =1, 2, 3, 4 .

a=2, l=2nand since there are nterms

the sum is

2 2

S n

a l

nn

nn

n n

2

2

2

21

1

n = +

= +

= +

= +

^

^

^

^

h

h

h

h

Use trial and error for dierent values of n.

When n=5, 25S6

55 65 = =^ ^h h . When n=10, S

6

510 11

3

275n = =^ ^h h .

When n=15, 200S6

515 1615 = =^ ^h h . This is sll not greater than 200so we increase n by 1again..

When n=16, S6

516 17

3

68020016 2= =

^ ^h h`16terms must be taken so the sum exceeds 200.

Here is an example where the value of nremains unspecied.


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L 15

Sequences and Series: Arithmetic Questions Knowing More

a

a

a

b

c

b

c

b

If these are arithmec sequences nd the value of the missing terms18.

Find the sum to 25terms of the arithmec series using the formula S n a n d2

2 1n = + -^ h6 @.

Find the sum of the series using the rst and last term formula, S n a l2

n = +^ h.

T6 25 3+ +

...32 30 28 26 24 22 20 18 16 14 12+ + + + + + + + + + +

...25 21 17 13 9 5 1 3 7- - - - - - - + + +

...5 3 1 1 3 5 7 9 11 13 15 17+ + - - - - - - - - - -

T20 42+ + c T T12 392 3+ + +

3 7 11 15 19 23 27 31 35 39 43+ + + + + + + + + +

9 16 23 30 37 44 51 58 65 72- - - - - - - - - -

1 10 19 28 37 45 54 63 72 81 90 99- - - - - - - - - - - -

19.

20.


18/42




L16


a

c d

b

Find the sum of these series.

An arithmec series has 20terms with rst and last terms 5and 195respecvely. Find the sum of the series.

...2 4 6 8 10 200.+ + + + + +

...1 0 1 2 3 4 199- + + + + + + + ...250 238 134+ + -

...93 87 81 6 3+ + + + +

21.

22.


19/42




L 17


a

b

c

A series has T 43 = and 12T T2 3+ = - . Find T2 .

The fourth term of an arithmec series is 28and the sum of the fourth and h terms is 32.

Find the h term.

Find the common dierence, d.

Using the formula for the nthterm, T a n d 1n = + -^ h with n=5and your answers from above, to nd

the value of a.

a

b

c

For the arithmec series ...1744 28 22 16 10 4+ + + + + +

Find the values of aand dand write down the nthterm Tn .

By using the last term, 4, and the formula for Tn , nd out how many terms are there in the series?

Find the sum of this series.

23.

24.

25.


20/42




L18


a

b

c

d

e

f

g

Find the sum of the rst 5mulples of 3, that is, 3 6 9 12 15+ + + + .

Is this an arithmec sequence? If so, nd d.

The sum of the rst nmulples of 3is an arithmec series and ...S n3 6 9 12 3n = + + + + +

Use the formula S n

a l2n = +^ hto nd the sum to nterms and show this sum is S n n23

1n = +^ h

Find the general term Tn .

26.

Find nif the sum is 315.

How many terms must be taken for the sum to exceed 560? (Use trial and error approach)

Why can this sum never be 170?


21/42




L 19

Sequences and Series: Arithmetic Using Our Knowledge

The starng salary of an accountant is $34 000per year. Aer each year of employment he will receive anincrease in the salary by an amount $1200.

a

a

What is his salary in the ninth year of employment?

Salary for rst year =$34 000

Salary for second year is: $34 000 + $1200

Salary for third year is: $34 000 + $1200 + $1200

$34000 $1200 2#= +

`Salary for ninth year is $34000 $1200 8#+

Or using the formula for arithmec series with:

$34000a= $1200d= n 9=

$ $

T a n d

T n

1

34000 1200 1

n

n

= + -

= + -

^

^

h

h

Therefore,

$34000 $1200 $43600T 9 19 = + - =^ h

How much are his total earnings for the rst nine years.

The total salary in the rst nine years is the sum to 9terms of an arithmec series. Denote the total

earnings to the end of the nthyear by Sn (sum to nterms). Since we know n 9= and the last term is

l 43600= then the formula for the sum to nterms of an arithmec series gives:

S n a l2

2

934 000 43600

349200

n = +

= +

=

^

^

h

h

Total earnings are $349 200

This is the last term in the series

Arithmec series have uses in the real world. The next example is an applicaon of arithmec series to a nancial

problem involving annual salary increases.

Applications of Arithmetic Series


22/42




L20


Here is another real world problem.

A secon of a stadium has 7seats in the rst row, 10seats in the second row, 13seats in the third row, and so on.

If the last row has 91seats, how many seats are in this secon?

This can be expressed as an arithmec sum as:

...

a

d

7 10 13 91

7

3

+ + + +

=

=

To nd the sum, the number of terms, n, is needed. The nthterm of this series is given by:

7 3

T a n d

n

n

1

1

3 4

n = + -

= + -

= +

^

^

h

h

The last term is 91.

Which term is this? Solve:

91

3 4 91

3 87

T

n

n

n 29

n =

+ =

=

=

So there are 29terms. The sum of the arithmec series using the formula is:

S n

a l

S

2

2

297 91

1421

29

29

= +

= +

=

^

^

h

h

So there are 1421seats in the secon.

First row, 7seats

Second row, 10seats

Third row, 13seats

Last row, 91seats. . .

.

.

.


23/42




L 21


...

...

2 1 20 2 2 20 2 3 20 2 32 20

40 80 120 160 1280

# # # # # # # #+ + + +

= + + + + +

This is an arithmec series.

The sum of the arithmec series is:

(last term)

40

40

32

1280

a

d

n

l

=

=

=

=

S n

a l2

2

3240 1280

21120

32 = +

= +

=

^

^

h

h

1 2 32

40m 640m20m

Emptying wheelbarrow

Going back to rell the wheelbarrow

Here is another more complicated real world problem.

A groundskeeper distributes ferlizer on the green using a wheelbarrow. Over several days he empes the

wheelbarrow in 20metre increments, and has to go back to rell each me. If he empes 32barrowfulls, how far

does he walk?

For the rst load he walks 20m and returns, so he has travelled m2 1 20# # . For the second load he walks

m2 20# and returns, so he has travelled m2 2 20# # and so on. In total he will travel

So the groundskeeper walks a total of 21 120metres.


24/42




L22


Here is an example relang to building.

a

b

Find the number of boards.

45

S n

a l

n

n

n

2

2702

4 8

12

2 270

n = +

= +

=

=

^

^

^

h

h

h

a=4, l=8

The dierence in length between adjacent boards is the common dierence d.

The last board is number 45.

T a n d 1n = + -^ h

m

cm ( d.p.)

4 8

.

.

d

d

45 1

11

10 091

9 09 2

` + - =

= =

=

^ h

4m 8mFloor

A oor in room has the shape of a trapezium as shown. It is to have oor boards put in. The dierence between

the lengths of adjacent boards is a constant and so the lengths of the boards form an arithmec sequence. The

shortest board is 4m in length and the longest board is 8m. The sum of the lengths of the boards is 270m.

Find the dierence in length between adjacent boards in cenmetres (correct to 2decimal places)


25/42




L 23

Sequences and Series: Arithmetic Questions Using Our Knowledge

The annual salary of a sales manager increases by $750each year, where in his rst year he earned $45 000.

a

b

What is his salary in the sixth year of employment.

How much money did he earn in total in the rst 5years of employment?

27.

28. Chairs in an amphitheatre are arranged in an increasing order where the number of chairs in a row is

increasing from 6unl 58. Find the number of chairs if the dierence in the number of chairs in rows next

to each other is 2.


26/42




L24

Sequences and Series: Arithmetic Questions Using Our Knowledge

An architect is nding the cost of building a mul storey building. The 1stoor costs $200 000, the 2ndoor

costs $225 000and each subsequent oor costs $25 000more to build than the oor below. What does it

cost to build a 40-storey building?

The temperature in a high pressured hot water tank is falling at a constant rate. A reading was taken each

10minutes and these were:

The nal reading taken was equal to 26c . How many readings were taken altogether.

, , , ...230 224 218c c c

29.

30.


27/42




L 25

Sequences and Series: Arithmetic Thinking More

The Greek sigma (/ ) symbol is used to write series easily. / is a short hand notaon for describing series in a

precise and concise way. Wring a sum in sigma notaon relies on nocing the paern in the series. In sigma

notaon, we have

T T T T T T n

n

1 2 3 4 5

1

5

= + + + +

=

/

The general expression for the nth term is Tn

Formula

Start

n=1 n=2 n=3 n=6 n=8

End

Write out the terms of the series given in sigma notaon as n2n 1

8

=

/ .

Write out the series in expanded form n4 16

n 0

+

=

/ .

The boom number, n 1= , tells you to start the rst term by substung in n 1= .

Then substute n 2= and add it on, then n 3= and add it on again, all the way up to n 8= .

Expanding the sigma notaon one gets

2 2 2 2 2 2 2 2 2n 1 2 3 4 5 6 7 8

2 4 6 8 10 12 14 16

n 1

8

= + + + + + + +

= + + + + + + +

=

^ ^ ^ ^ ^ ^ ^ ^h h h h h h h h/

Here is another example.

n4 1 4 0 1 4 1 1 4 2 1 4 3 1 4 4 1 4 5 1 4 6 1

1 5 9 13 17 21 25

n 0

6

+ = + + + + + + + + + + + + +

= + + + + + +

=

^^ ^^ ^^ ^^ ^^ ^^ ^^h h h h h h h h h h h h h h/

Sigma Notation

n2

n 1

8

=

/

The last value of nis n=5

The rst value of nis n=1


28/42




L26


Write out the series in expanded form 4 nn 3

2

-

=-

/ .

Write out in expanded form (without sigma notaon).

Write the series in sigma notaon: 5 11 17 23 29 35 41 47+ + + + + + +

n4 4 3 4 2 4 1 4 0 4 1 4 2

4 3 4 2 4 1 4 3 2

7 6 5 4 3 2

n 3

2

- = - - + - - + - - + - + - + -

= + + + + + + + +

= + + + + +

=-

^^ ^^ ^^ ^ ^ ^

^ ^ ^

hh hh hh h h h

h h h

/

Here is an example with negave numbers where more care is needed.

Here a two examples of wring the expanded form from sigma notaon.

a

b

n5 4n 2

6

+

=

/

n5 7n 3

5

- -

=

/

14 19 24 29 34

5 2 4 5 3 4 5 4 4 5 5 4 5 6 4

120

= + + + + + + + + +

= + + + +

=

^^ ^^ ^^ ^^ ^^h h h h h h h h h h

5 7 5 7

15 7 20 7 25 7

3 4 5 5 7

81

= - - + - - + - -

= - - - - - -

= -

^ ^ ^h h h6 6 6@ @ @

Some examples of wring a series in the sigma notaon follow.

This is an arithmec series with common dierence, d 11 5 6= - = , and a 5= .

Therefore the sum is

T n n5 6 1 6 1n` = + - = -^ h

n6 1n 1

8

-

=

/

Start at n=2and end at n=6

Start at n=3and end at n=5


29/42




L 27


Calculate the sum of the series: m7 4m 0

100

+

=

/

Write it out in expanded format

...7 4 4 11 18 704mm 0

100

+ = + + + +

=

/

This is an arithmec series,

, , ,4 7 704 101a d l n= = = =

S n

a d n

S

2 2 1

2

1012 4 7 101 1

35754

n

101 #

= + -

= + -

=

^^

^^

hh

hh

(number of terms)

The next example involves recognising that the series is arithmec and nding the sum.


30/42




L28

Sequences and Series: Arithmetic Questions Thinking More

Write out in expanded form (without sigma notaon) and evaluate

Write these series in sigma notaon

Calculate the sum of each of these series.

a

a

a b

b

b

j7j 0

5

=

/

6 12 18 24 30 36+ + + + +

j9 5j 1

7

+

=

/ n4 6n 90

233

-

=

/

k4 9k 1

8

-

=

/

2 9 16 23 30+ + + +

31.

32.

33.


31/42




L 29

Sequences and Series: Arithmetic More Questions

Basics

1.

2.

3.

What is the general term of the sequence 6, 8, 10, 12, ...?

What is the 3rdterm of the sequence with general term T n4 3n = + ?

a

a

a

a

b

b

b

b

c

c

c

c

d

d

d

d

T n2 4n = +

T n4n =

T n11 7n = -

2T n4n = +

T n3 5n = +

, , , , ...7

1

7

2

7

3

7

4

T n6 2n = +

T n9n = -

T n3 4 5n = - -^ h

T n6 8n = +

7 14T nn = - -

If the numbers 12,x, 24are in arithmec progression, what is the value ofx?

7

15

12

16

15

18

67

20

4. If T n19 9n = - , what is the value of T3?

-8 -7 7 8

5.

a

a

a

a

a

b

b

b

b

b

c

c

b

b

b

a

a

a

d

Which of the following does notrepresent an arithmec series?

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

For the sequences shown, write down the number of terms, and the values of T1, T3, T6.

Write down the rst 5terms of the sequence whose general term is given

What is the 5thterm of the sequence with nthterm

The nthterm of a sequence is given by T n8 5n = - . For which value of ndoes the n

thterm equal -162?

The nthterm of a sequence is given by T n11 29n = + . For which value of ndoes the n

thterm equal 260?

3, 6, 1, -7, 0, 45

2, 22, 42, ...

9, 6, 2, -3, -10, 20, 4

-6, -1, 4, 9, 14, 19, ...

-16, -8, T3, ...5, T2, 21, ...

-19, ...

-19, ...

7, ...

34, ...

Find the general term, Tn, of these sequence

What are the next three terms if the common dierence is 7?

What are the next three terms if the common dierence is -8?

What is the missing term if the common dierence is 8?

The fourth and eleventh terms of an arithmec sequence are 69and 167. Find the common dierence.

The rst term of an arithmec sequence is 19and the common dierence is -9. What is the seventh

term of the sequence?

Find the formula for the nthterm, Tn, of an arithmec sequence whose rst term, a, and common

dierence, d, are given by a=13and d=-8.

a b c d


32/42




L30


a

a

b

b c

c

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

Find a, d

20,x, 30

13, 20

FindT20

For the arithmec sequence 12, 25, 38, 51, ...

The eenth term of an arithmec sequence is 20T15 = . Using the formula T a n d 1n = + -^ h , and giventhat the rst term is a=188, nd the common dierence d.

The rst term of an arithmec sequence is -26and the eleventh term equals 64. Find the common dierence.

The common dierence of an arithmec sequence is 17and the tweneth term equals 347. Find the rst term.

The sixth and tenth terms of an arithmec sequence are given by T 76 =- and T 2710 =- . Find the general

term by substung into T a n d 1n = + -^ h , and solve a pair of simultaneous equaons for a, d.

Is 35a term of the sequence with general term T n76 4n = - ?

Which term of the sequence 5, 13, 21, 29, 37, 45, ...is 341?

Write down the numbers 18and 46and insert three numbers between them so as to give 5numbers in

arithmec progression.

The sum of the rst two terms of an arithmec sequence is 10and the sum of the next two terms is 18.

Find the rst term and common dierence.

If the sequence , ,a b c

1 1 1 is arithmec, show that ac b a c2 = +^ h.

Findxand the common dierence if the following are APs

Find the common dierence of the AP, then ndx, given that T 2510 = ;

If the sequence , , ,8 29T T2 3 is arithmec, what are the missing terms?

What is the sum of the series ...4 3 42- - - + ?

How many terms are there in the arithmec series ...5 7 9 11 31+ + + + + ?

Find the expression for the nthterm, Tn.

x+2, 10,x+8

13.25, 21.125

5x-2,x+5, 3x

15, 22 16, 24

7x-12, 7x-2, 7x+8

Knowing More

S2

47 4 4247 = - +^ h S2

46 4 4246 = - +^ h

65

92

S2

45 4 4245 = - +^ h S2

46 4 4247 = - - +^ h

32.

33.

13

-84

-27

The third term of an arithmec series is given by 28T3 = and the sum of 56T T3 4+ = - is given.

What is the fourth term T4?

If the sums 65S9 = and 92S8 = , what is T9?

14

-28

19

28

27

20

84

157

a b c d

a b c d

a b c d

a b c d

a b c d


33/42




L 31


a

a

a

a

a

a

a

b

b

b

b

b

b

d

e

f

g

c

c

d

c

c

c

c

b c

34.

35.

36.

37.

38.

39.

40.

41.

42.

If these are arithmec series nd the value of the missing terms

Find the sum to 25terms of the arithmec series using the formula S n a n d2

2 1n = + -^^ h h.

Find the sum of the series using the rst and last term formula, S n a l2

n = +^ h

Find the sum of these series by rst nding the number of terms, n, in the series using the last term and theformula T a n d 1n = + -^ h

An arithmec series has 41terms with rst and last terms -100and 140respecvely. Find the sum of the series.

The seventh term of an arithmec series is given by T 297 = and the sum of 905T T7 8+ = - is given.

What is the eight term T8?

The h term of an arithmec series is -1and the sum of the fourth and h terms is -15.

For the arithmec series 13+17+21+...+6613.

T23 50 3+ + T25 12- + + T T6 92 3+ + -

62+60+58+56+54+52+50+ ...

1+2+3+4+...+1000

Find the values of aand dand write down the nthterm Tn.

Find the sum of the rst 9mulples of 6, that is, 6+12+18+24+30+36+42+48+54.

-90-81-72-63+ ...

-4-9-14-19-24-29-34-39-44-49-54-59-64-69-74-79

By using the last term, 6613, and the formula for Tn, nd out how many terms are there in the series?

Is this an arithmec sequence? If so, nd d.

Use the formula S n a l2

n = +^ hto nd the sum to nterms and show this sum is 3S n n 1n = +^ h

Find nif the sum is 2610.

How many terms must be taken for the sum to exceed 7650? (Use trial and error approach, or logs)

Give a simple reason why can this sum never be 695?

3+6+9+12+...+1098

Find the fourth term.

-4-8-12-16-20-...-516

Find the common dierence, d.

-23-16-9-2+...+110

Using the formula for the nthterm, T a n d 1n = + -^ h and your answers from above, to nd the value of a.

-1022-1000-...-142

-1-2-3-4-5-6-7 + ...

50+29+8-13-34-55-76-97-118-139-160-181-202-223-244-265

Find the sum of this series.

The sum of the rst nmulples of 6is an arithmec series and Sn=6+12+18+24+...+6n.

Find the general term Tn.


34/42




L32


a

a

b

b

a

b

43.

49.

50.

51.

52.

53.

54.

55.

56.

57.

45.

46.

47.

48.

44.

37

4, 16, 64

$38 750

3, 27

2+3+...+40

What is his salary in the fourth year of employment?

What will be the cost of building the 9thoor?

How much money did he earn in total in the rst 7years of employment?

What will be the cost of building all 26oors?

4+6+...+44 -5-4-...+26 2+4+...+40

138

32, 64, 128

$40 000

1, 24

147

64, 128, 192

$41 250

1, 34

Find the sum of all posive mulples of 9which are less than 230.

Three numbersx,y,zare inserted between 0and 256, so as to give 5numbers in AP. What are the values of

x,y,z?

A man earns $30 000in his rst year at work, and gets an increase in his salary by $1250each year. What is

his salary in the 8thyear of his employment?

The sequencex, 7, 17,y are in arithmec progression. What are the numbersx,y?

Which of the following is an arithmec series with 21terms whose sum is 504?

The annual salary of a tradesman increases by $1250each year, where in his rst year he earned $32000.

An architect is nding the cost of building a mul storey car park. The rst oor costs $170 000, the second

oor costs $190 000, and $60 000for each addional oor. What is the cost of building a 12storey car park?

A new building has 26oors. The cost of building each oor varies. The rst oor costs $2 000 000. The cost

of building each subsequent oor will be $650 000more than the oor immediately below.

Find the sum of the integers between between 150and 250which are mulples of 7.

The sum of the rst 5terms of an arithmec series is -10, and the sum of the next two terms is -32. Find a, d.

Find the sum of the AP: ...2 3x x x nx+ + + + .

Find the sum: ...18 50 98 45 2+ + + +

Show that the sum to nterms of the AP 1+3+5+...is S nn2

=

Find the sum of the rst 100odd numbers.

114

16, 64, 128

$39 250

-3, 27

Using Our Knowledge

a b c d

a

a

b

b

c

c

d

d

a b c d

a b c d

The Chairs in a small amphitheatre are such that the rst row has 9chairs, and each row increases by 4unlthe last row, which has 33chairs. What is the total number of chairs.

Chairs in an amphitheatre are such that the rst row has 12chairs, the second row has 16chairs, and each

row increases by 4, unl the last row, which has 104chairs. Find the total number of chairs.


35/42




L 33


a

a

b

b

c

58.

59.

60.

61.

62.

63.

64.

65.

66.

67.

How far does Paula swim in the fourth week?

Find the number of planks.

In which week does she rst swim 4km?

Find the dierence in length between adjacent planks.

What is the total distance Paula swims in 30weeks?

Paula is training for a 4km swimming race by swimming each week for 30weeks. She swims 200m in the rst

week, and each week aer that she swims 200m more than the previous week, unl she reaches 4km in a

week. She then connues to swim 4km each week.

The temperature in a cool room was taken at regular intervals aer it was turned on, and the readings in

degrees Celsius were , , ,25 24.1 23.2 ...c c c . Assume that these readings are in arithmec progression. If the

nal reading taken was equal to .9 2 c- , how many readings were taken altogether?

What is the expanded form of the series m9m 1

6

=

/ ?

What is the expanded form of the series k3 3k

2

0

3

-

=/?

What is the sigma notaon for the series 4+7+10+13+16+19?

What is the sum of the series k3 3k 3

12

+

=

/ ?

Write out in expanded form (without sigma notaon) and evaluate

Write these series in sigma notaon.

Calculate the sum of each of these series.

Thinking More

a

c

b

a

a

b

b

9+18+...+54

3-1+9+24

1+9+...+54

-3+0+9+24

9+18+...+56

-3+0+3+6

1+18+...+54

0+9+24

n4 3n 1

6

+

=

/

S

2

912 399 = +^ h S

2

109 3610 = +^ h S

2

1012 3910 = +^ h

j4j 1

6

=

/

j5 9

j 4

10

-

=

/

j3j

3

0

5

=

/

n1

n 0

100

-

=

/

S

2

912 369 = +^ h

k3 2k 2

7

-

=

/ k3 40

7

+/ k3 2n 2

7

-

=

/

7+14+21+28+...+70

1000+999+998+997+996+995+...+1

-30-20-10-0+10+20+30+40

a

a

a

a

b

b

b

b

c

c

c

c

d

d

d

d

A tall fence has the shape of a trapezium and has planks arranged as shown. The dierence

between the lengths of adjacent planks is a constant and so the lengths of the planks

form an arithmec sequence. The shortest plank is 180cm in length and the longest

string is 250cm. The sum of the lengths of the planks is 774m


36/42


37/42


38/42

100% Sequences & Series



L36

Sequences and Series Answers

d=2cm

a b

36planksa

a

b

b

60.

61.

62.

63.

9+18+...+54a

-3+0+9+24b

k3 2k 2

7

-

=

/b

64.

65. 4 8 12 16 20 24j4 84j 1

6

= + + + + + =

=

/

j3 0 3 24 81 192 375 675j

3

0

5

= + + + + + =

=

/

66.

c

k7k 1

10

=

/ k10 40k 1

8

-

=

^ h/

k1000k 0

999

-

=

/

67. a b182S7 = S 4949101 =-

S2

1012 3910 = +^ hc

a b

c

58. 800m. 20thweek.

82km

59.39readings


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