21043-level 7 algebra.pdf

8/9/2019 21043-Level 7 algebra.pdf

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ALGEBRA

Algebra

Revision

Chapter

6 Inequalities

Chapter 7 Simultaneous Equations

Chapters

Sequences

Chapter 9 Algebra Review

9 2

107

119

141

159

91


2/75

Algebra

Revision

REVISION

DIVISIBILITY

A number

is divisible by 2 if it is an even

number

divisible

by

3 if the sum of its

digits

is divisible

by

3

divisible

by 4 if

the

number formed

from the

last

two

digits is

divisible by 4

divisible

by 5 if the last digit is 0 or 5

divisible

by 6 if it is divisible by both

2

and

3

divisible by 8 if the

number

formed

from

the last three digits is

divisible by

8

divisible

by

9

if the

sum

of its digits is divisible by 9

divisible

by

10 if

the

last

digit

is

0.

SPECIAL NUMBERS

A prime number is divisible by just two

numbers,

itself

and 1 .

The

first few

prime

numbers are 2,3,5,7,11,13.

The multiples

of a number are found by multiplying

the

number by each of 1,2,3,4, 5,

6,...

For

instance,

the first few multiples

of 5 are 5,10,15,20.

A

factor

of

a

given

number

is

a

number

that

divides

exactly

into

the

given

number.

For

instance, the factors of

20 are 1,2,4,5,10,20.

A square number is formed

when

a number is multiplied by

itself.

For

instance, since

2x2 =

4 then 4 is

a square number.

SEQUENCES

A sequence

is a

list of

numbers such as

3 ,

7,11,15,... There is

usually

a

relationship

between the numbers.

The

first

number of a sequence is called

the

first term,

the

second

number is called

the

second

term,

the

third

number

is called the third term,

and

so on .

Sequences

are sometimes based

on

the following

special numbers - odd

numbers,

even

numbers, squares,

cubes, multiples.

The terms of

a

sequence are sometimes found by adding the same

number

to each previous

term

or

by

multiplying each

previous term by

the

same

number.

For instance 1,4,9,16,... is a

sequence

of square numbers

2,5,8,11,...

is

a

sequence

in which

each

term

is 3

more

than

the

previous

term.

continued.

92


3/75

Algebra Revision

.

from previous page

The Fibonacci

Sequence

is 1,1,2,3,5,8,13,...

The

first

two

terms

are 1,1. Each

term

after

this

is found by

adding

the

two previous

terms.

Many sequences

can be found in Pascal's Triangle. The

first

few

rows

of this

are:

1

1 1

2

33

464

In

Pascal's triangle

the

numbers

down the

left

and right-hand sides are always 1 . That is,

each

row begins and ends with 1 . All

other

numbers

are

the sum of the

two

numbers

immediately above

on

the previous row.

EXPRESSIONS.

FORMULAE.

EQUATIONS

x + 3 is an expression.

a

= x

+ 3 is a

formula. The value

of a depends on the

value of

x.

Replacing

a letter

with a number is called substituting in a

formula.

For

instance,

if we are

told

x

=

6 in the formula a = x

+ 3,

then

replacing x

by

6 we

get

a

=

6

+

3

-9.

2p 4 = is an equation, p can have only

one

value; p

=

2-5.

SIMPLIFYING EXPRESSIONS

ab means a

x

b ba is the

same

as ab

2a means 2 x a

a 2 means a

x

a

5a +

2a

can be

simplified

to

7a

5a + 3b -

a

+ 2b can

be

simplified to

4a

+ 5b

When

we

remove

the brackets

from

5

(2a

-

3)

we

get

lOa

-

1 5 .

continued.

9 3


4/75

Algebra Revision

.. from

previous page

S O L V I N G EQUATIONS

Three methods

of

solving

equations

are: trial

and improvement, flowchart

method,

balance method.

The

flowchart

method

for

solving

2a

-

4

=

1

is

shown

below.

Begin

with

a

|x

2 | * - 2a >|_4| - 2a-4

2 - 5 EK 5 1+3

Begin

with

1

Hence

a

= 2-5.

The

balance

method for solving

2a-4=l

is shown

below.

2a-4

=

1

2a

=

5 (adding 4 to both

sides)

a

=

2-5

(dividing

both sides

by 2)

We

would take

these

steps to solve 2a -

4 =

1 using

trial

and improvement.

Guess a likely answer.

Check to

see

ifthis

answer is correct.

Make another guess and so

on.

The trial and improvement method

for

finding

the

solution

(to 1 d.p.) for the

equation 2x 3

-1=9 is shown

below.

Try x = 1 . Ifx = 1,2x

3

-1

=

1

which

is less than 9 .

Try x = 2. Ifx = 2,2x3 -1 = 1 5 which is greater than 9 .

Since

9

lies

between

1

and

15,

then

the solution must

be

between

1

and

2.

Tryx=

1-5.

Ifx =

l-5,2x

3 - =

5-75 which is less than 9 .

Tryx=

1 - 8 .

Ifx

= l-8,2x

3 -

=

10-664 which is greater than 9 .

Tryx = 1-7. Ifx = l-7,2x 3

-1

= 8-826 which is

less than 9 .

The solution lies between 1 - 7 and 1 - 8 . Since

8-826

is closer to

9

than is 10-664, the

solution to 1 d.p. is x

=

1 - 7 .

When

solving

an

equation always check your solution by

substituting

your s olution

back

into

the

equation.

For

instance,

to check that

p = 3 - 5

is the

solution

for

6p + 1

=

22 proceed as follows:

If

p

=

3 - 5

then

6p

+

1

=

6

x

3 - 5

+

1

= 22 Correct.

Take the following steps to solve a problem using equations.

Step

1

Choose

a variable

such as n or x for the unknown q uantity.

Step

2

Rewrite the

statements

in mathematical symbols.

Step 3

Combine

these statements into an equation.

Step 4 Solve the

equation.

Step 5 Check the answer with the

information

in the

problem.

continued...

94


5/75

Algebra Revision

. from

previous

page

COORDINATES. DRAWING a LINE

The x-axis

is the horizontal axis.

The y-axis is

the

vertical

axis.

The

coordinates ofa point are

a

pair of

numbers

such

as (3, -2) .

The

first

number is the

x-coordinate;

the second

number

is the

y-coordinate.

For the

point

P (3, -2), the x-coordinate

is

3

and the

y-coordinate

is

-2.

The graph

of

a straight line may be drawn as

follows.

Step

Find the coordinates

of three

points on

the

line.

Step 2 Plot these

points.

Step 3 Draw the line that passes

through

these points.

Note: The line could

be

drawn by plotting just two

points

but for

greater

accuracy

it is

wise to plot three

points.

For instance, to draw the line y = 2x

+

1 proceed as follows.

Choose three

values forx,say-l,0,1.

Subs titute these values

for

x into y

= 2x+l

to

find the

corresponding

values ofy;

see

the table below. Now plot the

points (-1,

-1),

(0, ),( , 3 ) and draw the line that goes

through

these points;

see

the graph below.

X

y

-i

-i

0

i

i

3

= 2x

fl

The lines y

=

2x, y

= 2x + 1 ,

y

= 2x + 5

etc. are all

parallel since

the number

multiplying

the x

is

the same for

all of

them.

The

lines

y

=

x

+

4, y

=

2x

+

4, y

=

5x

+

4 etc. all

meet

the y-axis

at the

same place

since

the number

added

is

the

same for

all

of

them.

9 5


6/75

Algebra Revision

REVISION

EXERCISE

1. Find the

missing

term in these sequences.

(a)

1, D, 9,16,25,...

(b)

1,2,

D,

8,16,...

(c)

1, 8 , D,

64,125,

(d)

1,8,0,22,29,...

2.

65742 54627 46725

3 6472 4723 6

56664

673 48 72654

Which number in this list is divisible by both 3 and 8 ?

3 . (a) John's lucky number is n.

Daniel's

lucky

number

is twice

John's.

Beverley's lucky

number

is three

less

than Daniel's.

Write

an express ion for Beverley's lucky

number.

(b)

Five

added

to

twice Beverley's lucky number is 23.

W rite and

solve an equation to

find

Beverley's

lucky

number.

4.

Stacy is

baking chocolate

cakes

for the

school fair.

The

first

one took

her 35

minutes.

The

next ones each took 20 minutes.

35,55,75 are

the first

3

terms

of

the

sequence that gives the total time

Stacy

took.

(a)

What is

the

sixth term of this

sequence?

(b)

Stacy

had allowed 4 hours

to

bake

these

cakes,

How many could she bake in

this

time?

5.

Simplify

the following.

(a) ax

a

(b) a +

a

(c) 3n-2n

(d)

4n-n-n

(e) 2a x 5b (f) 5a-3n-a + 2n

96


7/75

Algebra Revision

6 .

Use

a method

of

your

choice to

solve

these equations.

(a) 2n = 1 5

(d) 7n-2n =

(g) 3(n-7) +

(b)

2 + n =

1 5

(e) 7n-3 = 5n

(c) 4(2n-3) = 1 5

(f)

--1 = 5

= n

7 . Use

this graph

to answer

the following

questions.

(a)

The

speed

limit through

a

village

is

40mph. What is

this in km/h?

(b)

How

many

miles

per

hour is a speed

of

40km/h equivalent

to?

(c)

Sally

and

Rex both left Forth

at the

same time.

They travelled along the

B4277

to Tylorstown.

Sally

averaged

48mph; Rex

averaged 78 km/h.

Who arrived

at

Tylorstown

first?

Speed Graph

:-4-

+

75 ..... .3:.. ....f-

70:::-::::::::::::::::::

. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

6 5 ................33:

60

--

--,

55 ::-::::::B

50 .I........:.:...:

2 : ^::~:::::

:::

::::

: : : : :

|

5 .....|...

............

1* .__ _

_,_

__.---,

__--,

-_-

p , 40 .......... .........

w

_____

_ :r_ _ _

.--

...i..x____

-u ic - ~ - - - v

s - - - - - - ------ - ~ A

O -.-

..._-

------- ^ .. - X

3 3 0

::::::::::::::::-

25

....T........ ......

20 .

-.|--|--,-

7 _ x

15 -V---r-

0 .....i... ...+......

I.

5 : :-:::::::x::::::::

- f - M

"

^

z .

in

i^

?n

_

._

-_.___r.__ _ .

__.______..

______L-____t___,_

.-,I-T:_--_.._-_-_--_. Z-__.-

.._._-_--._-..---_._..---../.-.-..-__

-_,_-. __

^

_^ .-_

L

J_L __--.____,.____

"----

-------

- - --

'

I

_/-L_--

J_-_-

---.--_j_--_ --._---

^ ------ - i [

--

----- -_--_ --__-

i 1

;: ::iEll|i ^ :::

:_:___: _._

_____:_.._

- 1 i j |

[1

7 ? ^n * < ; dn .i s

miles

per

hour

8. Write down the first 5 terms of these sequences.

(a)

First term

equal

to

1 .

Every term

is

3 more than the previous term.

(b) First

term

2.

Every

term is

twice

the

previous term.

9 .

(a)

Remove

the

brackets

3

(4n

-

7 )

(b) Remove the brackets and simplify

3

(4n

- 7 ) + 2 (3

+

2n)

97


8/75

Algebra Revision

10.

Drawup

a set of

axes. Label

the x-axis from-6 to 10; label the y-axis

from-5

to 5.

Plot each point in the following

lists.

Join the

points

in order, as you

plot

them.

(a)

(5,2),

(5,

-4)

(b) (-5,2), (-1,4),

(9 ,4 ) (c) (-5,

-4),

(-1,

-2)

(d)

(-1,4),

(-1,

-2),

(9,

-2),

(9 ,4) , (5,2) ,

(-5,2),

(-5,

-4),

(5,

-

4), (9,

-2)

What

do you get?

11.

(a)

I am

a

factor of 36.

I am not a

multiple

of4.

I

am not

a square

number.

I am not a prime number.

I

have

one

digit.

What number am I?

(b) I am

a

factor of36.

I am a multiple of

3 .

The sum ofmy two digits is a square.

The

product ofmy two digits is

a

cube.

What number

am

I?

12. C =

299

+

18d

+

5 (d

+ 3) .

This formula

gives

the

cost

ofhiring

a

car for 7 days or longer.

C is the

total cost

in .

d is the number

of

days over 7 .

Find

the cost

of

hiring one

of these

cars

for 1 1

days.

13.

In words,

give

a rule that

generates each

of hese

sequences.

Test the rule, then

write

down the

next three

terms ofeach

sequence.

(a) 500,50,5,0-5,...

(d)

3,12,27,48,...

(b)

3,6,12,24,...

(e)

1,1,2,3,5,8,...

14. Each of4

classrooms

contained the

same number of students.

When 3

students left

one

of hese

rooms

there

were

77

students left.

Write

an

equation

for the number

of

students

originally

in each classroom.

(L et this number

be n. )

(c) 3,6,9,12,,

98


9/75

m

P

0 40

80

100

Algebra

Revision

15. Marks (m) out of

8 0 can

be changed to

percentages (p) by

us ing the

relationship

p

= l-25m.

(a) Copy

and

complete

this table.

(b)

Copy and

complete these coordinates.

(0, ), (40,

), (80,100) .

(c) Draw the graph of p = l-25m

for

m between 0 and 80.

(d)

O n a test that was marked out of

80,

three students were given the following

marks:

Tamara -76, Timothy-28,

Tewfik -45.

Use

the graph

you

drew

in

(c) to find the percentage marks

for

these students.

(Answers to the nearest % ).

16.

Neroli,

Laxmi,

Sara and

Nabila

all

play

basketball.

In one

match,

Neroli scored

x goals;

Laxmi

scored

3

more than Neroli, Sara scored twice

the number

L axmi did and

Nabila

scored 3

less than double the number Neroli did.

Write expressions for the number of goals

Laxmi,

Sara and

Nabila scored.

17. Find the coordinates of three points on the graph of y = 2x -1.

Draw

the graph

of

the

line

y = 2x-l.

18.

C ontinue the sequence 8,13,18,...

in

two different

ways.

1 9 .

3x l

x + 5

(a) Write and solve

an

equation

to

find x.

(b) Find the

length of this rectangle.

9 9


10/75

Algebra Revision

20. (a)

Helen was

using "trial

and improvement"

to solve the

equation 2x 3 =

40.

She wrote down this

table

ofvalues.

Explain why there is a solution to the

equation 2x 3 = 40 between

x

= 2

and x = 3 .

X

2x 3

i

2

2

1 6

3

54

4

128

5

25 0

(b)

2 x

2 0

1 6

2 1

1 8 - 5 2 2

2 2

2 1 - 2 9 6

2 3

2 4 - 3 3 4

2 4

2 7 - 6 4 8

2 5

3 1 - 2 5

2 6

3 5 - 1 5 2

2 7

3 9 - 3 6 6

2 8

4 3 - 9 0 4

2 9

4 8 - 7 7 8

Use

this

table

to

give the

solution

to

2x 3

=

40 accurate to

one

decimal

place.

(c)

Find

the solution to

2x

3 = 40 accurate to 2 decimal places.

(d)

What number

goes

in

the

gap

in

the

following

statement?

"The

equation

x

2

+

3x

=

10

has

at most __solutions."

(e) Use "trial and

improvement"

to find all the solutions

for

the

equation

x

2 +

3x = 10.

21. Write a program

to print the first

100 terms of

the sequence 20,24,28,32,...

EXAM.

QUESTIONS

22.

Write

the

missing

numbers in

these

simple

sequences,

(i) 4 7

10

13 .... .... 22

(ii)

1

2

4 8 1 6 .... 64 1 2 8

U L E A C

23 . (a) Plot the points (1 ,3) and (4,6) .

Join the points w ith a straight

line.

(b) The

point

P

(a,

5) lies

on

the line.

What

is

the

value of

a?

SEG

24.

Write

down

a

simplified expression

for

the

perimeter

ofeach of hese shapes.

a ,

M 4a

.

b)

i

| i

b

(a)

2a

b

2a

M EG

100


11/75

Algebra

Revision

25. The

statement

'All

multiples of ive

end

in

a

five' is not true all the

time.

Write down a statement

about

the

multiples of

five

which

is true all the

time.

M E G

26 . (a )

When

a number is multiplied by 4

and

3 is added to the result, the answer is 31.

What is the

number?

(b)

When

a

number

is

divided

by

4

and

3

is

taken

away

from

the

result,

the

answer

is

21.

What is the

number? W J E C

27 . Using the rule

double

the previous

number

and add

one,

write down

the

next three terms of the

sequence

1,3,7,

SEG

28 .

Row A

RowB

RowC

3

9

5

25

7

49

(a ) (i)

Write

down the

next two numbers in

row B.

(ii)

Explain

how

you

got

these

answers.

(b)

(i) Write down the

next

two

numbers in

row

C.

(ii) What is the name

given to

the type of numbers in rowC?

(c)

Complete this sentence by

putting

the

correct word

in

the space.

Iand7 are.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . of 49 .

SEG

29 . Solve the equation 6x +

25

= 97 .

ULEAC

30 . Robert is

investigating matchstick

patterns. The diagram

below

shows

three

patterns that

he

made.

UJ[

square

4 matchsticks

2

squares

7 matchsticks

3 squares

10 matchsticks

Which of the following formulae is

correct for

calculating the number of matchsticks M

needed

to make

a

pattern

with 5 squares?

Show the working you do to check your answer.

(i) M

=

S 3

(ii)

M =

2S

3

(iii) M

=

3S

+ 1 MEG

101


12/75

Algebra Revision

31. Look

at

this

pattern

Linel

1 + ( 1 X 1 )

=

Line

2

2 + (2 X 2) =

Line

3 3 + (3 x 3 )

=

Line4

4

+ (4

x

4)

=|

(a) Fill in the miss ing numbers in the

boxes.

(b) Write down Line 5 of

this

pattern.

(c) W rite down Line

9 9

ofthis pattern.

1x2

2x3

3 x

NEAB

32 . Jennifer does

an

experiment with

a balance.

She finds

that

three

packets ofbiscuits plus

100

grams weigh

the same

as

two

packets of

biscuits plus

250 grams.

A packet ofbiscuits weights

x

grams.

(a)

Form

an equation in

x.

(b) Calculate the weight of

a

packet ofbiscuits.

33 . Solve

these

equations,

(a) 3x + 2 = 18-5x

(b)2(x

+ 3)=

18-6*

SE G

SEG

34 .

Trevor

is making patterns

with

matchsticks.

Pattern

1 Pattern 2

Pattern

3

Pattern

4

(a)

Complete

this

table.

Pattern

1

2

3

4

5

Number of

matches

3

Number o f

T r i a n g l e s

1

(b) How

many

matches are needed for pattern

12 ?

(c) Describe the

connection between

the number

of

matches and the

number

of triangles.

M E G

102


13/75

Algebra Revision

35 . Sarah and

Owen

do

not yet know how many people are

coming

to

the party.

[How many

sandwiches

will

If each

person

eats

4

sandwiches, we multiply the

number

of

people by

4.

I

W e had better make 6

extra

ones as well,

however many

people

there

are.

(a)

Sarah

works

out that if there are 3 people at

the

party,

they

need to make

18

sandwiches.

(i) Write down

Sarah's

calculation,

(ii)

Complete

the

table.

Number of

people,/)

Number

of

sandwiches,

s

1

1 0

2

14

3

1 8

4

5

6 7

8 9

42

1 0

(b) Write down a formula

to

work out

how

many

sandwiches,

s, they make if p

people

come to the

party.

(c ) Owen uses

a

formula to work out how

many

sausage rolls,

r,

to make if p people come

to the

party.

He

uses

r

= 2p

+

5.

(i) Use Owen ' s formula to work

out

how many sausage rolls he needs to make if 3 0

people come

to the party.

(ii)

He works out that he needs

41

sausage

rolls.

How many

people

is he

expecting? NEAB

36. (a) Complete this

table

of

values for y

= x 3.

X

y =

x 3

-3

-6

-2

-5

-1

-4

0

-3

1

-2

2

3

4

5

(b)

Plot the points

and

draw

the graph

for y = x 3 .

U L E A C

3 7 . Michael

has

a set of

play

cubes.

Each

cube

has

the numbers

1 to

6

printed on

the

sides, as shown.

(a )

Which of

these

numbers, greater

than 1, are

prime

numbers?

103

SEG


14/75

Algebra

Revision

(b) Michael tries to use the cubes to form

a

pattern of square numbers up to 50.

Two

numbers in

the pattern

are

missing.

I

What

are

the

missing

numbers?

(c) Michael

uses

the cubes to start another

number

pattern.

(i)

What is the next

number

in

this

pattern?

(ii)

The

number

pattern

is

continued.

Explain

how you would

find the seventh

number

in

the

pattern.

SE G

8

100

kilometres

9 0

70

60

50

4 0

30

20

10

10

20

30

40

50 60 70

miles

104


15/75

Algebra

Revision

The

conversion

graph

opposite can be used

to

convert between miles and kilometres.

Use the

graph

to

help

you answer the questions,

giving

your answers to the nearest whole

number.

(a) Convert 50km to

miles.

(b) Convert 27 miles to km.

(c) Explain how the information on the graph could

be

used to

convert

10 000km to miles.

ULEAC

39 .

G eorge has to

find

a

solution

to

the equation

x 2 + 2x

= 10,

correct to one decimal place.

First

he

tries x =

3-0

and

finds

that the

value of

x 2

+

2x

is

15.

By trying other values

of

x

find

a solution to the equation

x 2

+

2x

= 10, correct to one

decimal

place.

You must show

all

your working. S E G

40. (a) For the

equation

y y +4, choose

three

values of x in the range 0 to

60 and

work

out

the

values

of

y.

(b) Draw the graph of y = y +4

for values

of

x from 0 to 60.

This graph shows the relationship

between

the

perimeter,

x inches, of a picture frame and

the

cost, y.

A frame costs 14.

(c) Use

the graph

to

find

its

perimeter.

M E G

4 1 .

fig 2

fig

A

clothes

dryer

is

in the shape

of

an upside down, square based pyramid

(fig

1 ) . It has

three

strands

of line along each

side of

the pyramid

(fig 2). The s trand

AB is x feet long.

DC is

2

feet

shorter than

AB.

EF is 2

feet

shorter than DC .

(a) (i) Write the length of DC in

terms

of x.

(ii) W rite the length of EF in

terms

of x.

(b)

What

is

the total length of

line needed for

the

clothes dryer

in

terms

of x?

(c) The total length of line

needed is

48

feet.

What

is

the length of

AB?

N I C C E A

1 0 5


16/75


17/75

D

Inequalities

NOTATION for

INEQUALITIES

Remember:

< means "is less than"

means

"is greater

than"

> means

"is greater than or

equal

to"

n > -5

is

read as "n

is greater

than

-5"

n

>

-5 is

read

as

"n

is greater

than or

equal

to -5"

n < 3 is read as "n

is

less than

3 "

n < 3 is

read

as "n is less

than

or

equal

to 3 "

-

4

< n < 7 is

read

as "n is between -

4

and 7 "

or as

"n

is

greater than

-

4 but less than 7"

-

4

60

T-bone

steaks weigh

between lOOg and 200g.

100


18/75

Inequalities

Hans never arrives at school

earlier

than 8-30a.ni.

t

8-30

t > 8-30

Shane

takes

between 3 and 4

minutes to iron

a shirt.

3

35

In a test, every student

gained

at least 70 % .

m

< 7 0 %

m

70%

m >

7 0%

Write

down some

other

statements

that could be described

by

inequalities. Use

inequalities

to

describe

them.

Discuss.

DISPLAYING INEQUALITIES on the N U M B E R

LINE

DISCUSSION EXERCISE

6 :2

If

n +

4

=

6,

n can have

only

one value.

What

is

this

one value?

Which whole number values could n have

if

n

+ 4 >

6?

Is 2-5 also a solution for n +

4

> 6? Is 2-1? Is 2\ ?

Can you list all

the solutions

for n

+

4 > 6?

Discuss.

Which

of the

following

show

all

the solutions for n + 4 > 6?

Discuss.

What meaning

could

be attached to

the

symbols and

o?

What

meaning could

be attached to

the

arrow?

Discuss.

-i

-l

To display an inequality on the number

line proceed

as follows.

Step

1

Draw

a

line over all the values included.

Step

2

If the end point

of

the line

is one

of

the values included, place the

symbol

on

this end

point; if the end point is not one of the values

included,

place the

symbol o on this end po int.

108


19/75

Chapter

6

W o r k e d Example

Display

these on the number

line.

(a) x

>

-2 (b) a < 3 (c)

1

2 '34--- 3 ' 2' 1 '

4'

5''

' '

l> 3 '5> 7 >... 3'9'2'15'-'-

INVESTIGATION

8 :12

EXPLORING SEQUENCES using the COMPUTER

10 FORN

= 1TO5

20 T = (N + 1)/N

3 0 PRINT

T

40

NEXT

N

50 END

Type this

program into

a computer. What sequence is generated? Adjust

the

program at

line

10

to

print

more

terms

of

the sequence.

Investigate

questions

such

as

:

How

many

terms need to be generated to find a term which is less than 1 - 0 1 ?

Adjust the above

program

at

line 20

to

generate

other

sequences.

You

could

consider

t n

=

n + 1 , t n

,

t < ~

n

J , t n = 3 n 2 ,

i , , =

1

-n.

Explore

these sequences.

E XA M .

QUESTIONS

1.

The first five terms of

a simple

number

sequence

are

5 8 1 1 14 17

Find,

in terms of

n, the nth

term of

the sequence. ULEAC

2. Consider

the

sequence

1 , 5, 9, 13, 17,

21, 25,

. . . . . .

(a)

Find

the next term

in

the

sequence and exp lain how

you

obtained

your answer.

(b)

The

nth term

in

the

sequence

is

4n

3 .

Solve the equation 4n =

3 97 and

explain what the

answer tells

you. MEG

155


66/75

Sequences

3 . Here are the first four terms ofa

number sequence,

7, 11, 15, 19.

Write

down

the nth term

of

the

sequence.

4 .

On this

piece

of

bunting

there

are

three light

flags

between

every

pair of

dark

flags.

All

the flags

are

numbered

in order.

The first dark flag is

number 1,

the second dark flag is number.

5

and so on .

ULEAC

1234567

(a) Complete

this

table

by filling in the numbers

of

the

dark flags.

Dark flag

Flag number

1s t

1

2nd

5

3 rd 4 th 5th

10th

(b) Find an expression for the number of

the

nth

dark

flag.

N E A B

5. The three

patterns

below are made out ofmatchsticks.

Pattern Pattern 2 Pattern 3

(a)

Draw the

next pattern in

the sequence.

(b)

Complete this

table

to

show the

number ofmatchsticks used for

each pattern.

Pattern

number

Number of matchsticks

1

4

2

1 0

3

1 6

4

5

6

(c)

How

many matchsticks

would be

needed

for

the 20th

pattern?

Show

clearly how you

worked

out

your

answer.

(d) Write

down

an expression for

the number of

matchsticks

in

the nth

pattern. NEAB

6 . Write

down

the

next number and a

rule

for continuing each ofthe following number

patterns.

(a)2,7,12,17, .............

(b)

1,3,9,27, ............. WJEC

156


67/75

Chapter 8

7 . Write

the next two numbers in

each

of

these simple sequences.

Write

a rule for finding

the

next

number

in

each

sequence.

(i) 4

(ii) 3 0

(iii)

1

8

25

4

12

20

9

16

15

16

U LEA C

8 .

(a)

Write down

in terms

of n, the nth term

of the

sequence

3, 7,

11,

15, . . . . . .

(b) Write down

in terms

of n, the nth term

of the

sequence

9. A

sequence

begins

1

1

J_

_L

3 '

9 '

27

'

8 1

'

2, 2,

5,

13,

28 ,

SEG

The numbers

in

the

sequence

can be

calculated using

differences.

In each case

the

numbers

in the

rows

A

andB are

the

differences

of the

numbers above them.

Sequence

Row

A

RowB

2 2 5 13 28

0 3 8 15

.

3 5 7

. . . . . . .

By

continuing

row

B

and

then

row

A

write in

the

next

two numbers

in

the

sequence. SEG

10.

(a)

The

first

three

terms

of

a

sequence are

349

> ,

t,

7 ,

The diagram shows

how

to find each term in

this

sequence.

9

i)

Write

down

the

fourth term ofthis

sequence,

(ii) Calculate

the sixth term of this

sequence.

(b)

Find

the next term in

each

ofthe

following sequences.

Show clearly how you got your answers.

(i)

4,

10,

19, 31,

46,

. . .

(ii) 243, 81, 27, 9 , 3,

(c )

Write

down

the nth

term

for each of the following

sequences,

(i)

1 ,

4,

9,

16,

25,

. . .

(ii)

3, 5, 7 ,

9 ,

11, . . .

157

NEAB


68/75

Sequences

11.

Look

at this sequence

28

18 3 2

(a) Write

down

the next

term

of the

sequence.

(b) Write down the 25th term of the

sequence.

(c)

Write

down

the

nth

term of the

sequence.

12. Give the nth term of

these

sequences.

1st

term 2nd term 3rd term 4th term

a)

(b)

13. Jayant and

Paula

make patterns like this from

matchsticks.

M E G

5 7 9 1 1

1st

term

2nd

term 3rd term

4th

term

0

3 8

1 5

SEG

Number

1

Number

2

Number

3

Here is

their

table of results.

Number

4

Pattern number

(n )

Number of small triangles

in

pattern (t)

Number of matchsticks

needed

(m )

1

1

3

2

4

9

3

9

1 8

4

30

5

6 7

8

Complete the table for them.

14. Write down the

rule

for each of these sequences.

(a)

3-4, 3-0, 2-6, 2-2, 1-8,

. . . . . . . . . . . . . . . . . . .

(b)

3 , 6, 12, 24, 48,

. . . . . . . . . . . . . . . . . . . .

(c) 3 ,

4,

6, 9, 13. . . . . . . . . . . . . . . . . . . . .

N E A B

SE G

158


69/75

Algebra Review

1. Use either

the

balance method or

the

substitution

method

to solve

these

simultaneous

equations. (a) 5x

- 6y

=

8

(b)

x - 2y -

1 = 0

4x + 3 y = -17 x

+

4y = 4

2 . A

sequence

is given by

the

rule

r , ,

=

2n

- 1 .

(a) W rite down

the

first

five terms of

this

sequence,

(b) What is the

91s t

term?

(c)

Which

term

is equal

to

121?

3.

x +

y

= 5

2x 3y 3

13 11

(a) Use

the graph to find the solution of the simultaneous equations 2x - 3 y = 3 and

(b) Rearrange y = -x

+ 5

as

x

+

...

=5.

(c)

Use

the graph to

find

the values of

x and

y for which y -x + 5

and

y = - \ x + 6.

(d) Use

the

graph

to

write

down two

simultaneous equations which

have the

solution

x = 3 , y = 2.

159


70/75

Algebra

Review

4 . (a)

"Last

night the

temperature did not fall below 6C." Which

of the following

inequalities describes this

statement?

A. T 6

C. T 6

(b) "No worker at

A.B.

Tyres earns less than 4 an hour

and no

worker

earns more than 9

an hour." Which of the following inequalities best

describes

this

statement?

A.

E>4

B. E


71/75

Chapter 9

7 .

Daley decides to save

2

this week, 5 next week,

8

the week after and so

on;

so that

in

one

week

he

will

save 3 more

than in the

previous week.

If

Daley keeps to

this

savings

plan,

how

much

will

he

save

in the

27 th

week?

8 .

Find

the

next

two

terms

of

these

sequences.

You

may wish

to

use

the

difference

method.

(a)l,6,ll,16,... (b) 5,18,37,62,93,... (c)

2,11, 32,

71,134 ,

227 , .

..

9 .

Choose

the

correct

inequality for each

graph.

( a )

- 3 r

A.

n> 1

B. n< 1 C.

n

>

1 D. n -l B . n

-1

c)

210123

A. -1

21043-level 7 algebra.pdf

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