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SAMPLE
CHAPTER
INTERNATIONAL
GCSE
(9–1)
ALAN SMITH
SOPHIE GOLDIE
Mathematics
for Edexcel Specification A
THIRD
EDITION
1
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INTERNATIONAL
GCSE
(9–1)
ALAN SMITH
SOPHIE GOLDIE
Mathematics
for Edexcel Specification A
THIRD
EDITION
Contents
1 Fractions, decimals and rounding
Starter: Half and half
1.1 Equivalent fractions
1.2 Multiplying and dividing with
fractions
1.3 Decimals and fractions
1.4 Rounding and approximation
1.5 Rounding calculator answers
1.6 Upper and lower bounds
Key points
Internet Challenge 1
2 Ratios and percentages
Starter: How many per cent?
2.1 Working with ratios
2.2 Simple percentages
2.3 Percentage increase and decrease
2.4 Reverse percentage problems
2.5 Compound interest
Key points
Internet Challenge 2
3 Powers and roots
Starter: Roman numerals
3.1 Basic powers and roots
3.2 Higher powers and roots
3.3 Fractional (rational) indices
3.4 Negative powers
3.5 The laws of indices
3.6 Standard index form
3.7 Calculating with numbers in
standard form
3.8 Factors, multiples and primes
3.9 Highest common factor, HCF
3.10 Lowest common multiple
Key points
Internet Challenge 3
4 Working with algebra
Starter: Right or wrong?
4.1 Substituting numbers into formulae
and expressions
4.2 Working with indices
4.3 Expanding brackets
4
4.4
4.5
4.6
4.7
Multiplying two brackets together
Factorising – common factors
Factorising – quadratic expressions
Factorising – harder quadratic
expressions
4.8 Factorising – difference of two squares
4.9 Generating formulae
4.10 Changing the subject of a formula
Key points
Internet Challenge 4
5 Algebraic equations
Starter: Triangular arithmagons
5.1 Expressions, equations and
identities
5.2 Simple equations
5.3 Harder linear equations
5.4 Equations and brackets
5.5 Equations with fractional
coefficients
Key points
Internet Challenge 5
6 Graphs of straight lines
Starter: Matchstick puzzles
6.1 Coordinates in all four quadrants
6.2 Graphs of linear functions
6.3 Gradient and intercept of linear
functions
6.4 Equations and graphs
6.5 Parallel and perpendicular lines
Key points
Internet Challenge 6
7 Simultaneous equations
Starter: Fruity numbers
7.1 Solving simultaneous equations by
inspection
7.2 Solving simultaneous equations by
algebraic elimination
7.3 Solving simultaneous equations by
a graphical method
7.4 Setting up and solving problems
using simultaneous equations
Contents
Key points
Internet Challenge 7
8Inequalities
Starter: Treasure hunt
8.1 Whole-number solutions to
inequalities
8.2 Using algebra to solve linear
inequalities
8.3 Illustrating inequalities on a
number line
8.4 Graphs of linear inequalities in
two variables
8.5 Quadratic inequalities
Key points
Internet Challenge 8
9 S Number sequences and seies
Starter: Circles, lines and regions
9.1 Number sequences
9.2 Describing number sequences
with rules
9.3 Arithmetic sequences
9.4 Arithmetic series
Key points
Internet Challenge 9
10Travel and other graphs
Starter: Animal races
10.1 Distance–time graphs
10.2 Modelling with graphs
Key points
Internet Challenge 10
11Working with shape and space
Starter: Alphabet soup
11.1 Corresponding and alternate angles
11.2 Angles in triangles and quadrilaterals
11.3 Angles in polygons
11.4 Areas and perimeters of simple
shapes
11.5 Surface area and volume
Key points
Internet Challenge 11
12Circles, cylinders, cones and spheres
Starter: Three and a bit …
12.1 Circumference and area of a circle
12.2 Sectors of a circle
12.3 Circumference and area in reverse
12.4 Surface area and volume of a cylinder
12.5 Exact calculations using pi
12.6 Volume and surface area of cones
and spheres
Key points
Internet Challenge 12
13Geometric constructions
Starter: Round and round in circles
13.1 Constructing triangles from given
information
13.2 Constructions with line segments
13.3Bearings
Key points
Internet Challenge 13
14Transformation and similarity
Starter: Monkey business
14.1Reflections
14.2Rotations
14.3 Combining transformations
14.4Enlargements
14.5 Similar shapes and solids
Key points
Internet Challenge 14
15Pythagoras’ theorem
Starter: F
inding squares and square
roots on your calculator
15.1 Introducing Pythagoras’ theorem
15.2 Using Pythagoras’ theorem to
find a hypotenuse
15.3 Using Pythagoras’ theorem to
find one of the shorter sides
15.4 Pythagoras’ theorem in three
dimensions
15.5 Pythagoras’ theorem on a
coordinate grid
Key points
Internet Challenge 15
16Introducing trigonometry
Starter: A triangular spiral
16.1 The sine ratio
16.2 The cosine ratio
5
Contents
16.3 The tangent ratio
16.4 Choosing the right trigonometrical
function
16.5 Finding an unknown angle
16.6 Multi-stage problems
16.7 Angles of elevation and depression
Key points
Internet Challenge 16
17Circle theorems
Starter: Circle vocabulary
17.1 Tangents, chords and circles
17.2 Angle properties inside a circle
17.3 Further circle theorems
19.4 Intersecting chords
Key points
Internet Challenge 17
18Sets
Starter: Does it all add up?
18.1 Introducing set notation
18.2 Venn diagrams
18.3 Further Venn diagrams
Key points
Internet Challenge 18
19Working with data
Starter: Lies, damned lies and statistics
19.1 Calculations with frequency tables
19.2 Solving problems involving the mean
19.3Histograms
19.4 Cumulative frequency
19.5 Median and quartiles for a
discrete data set
Key points
Internet Challenge 19
20Probability
Starter: Dice throws
20.1 Theoretical and experimental probability
20.2 Mutually exclusive outcomes
20.3 Independent events
20.4 Tree diagrams
Key points
Internet Challenge 20
21Direct and inverse proportion
Starter: A sense of proportion
21.1 Direct proportion
21.2 Inverse proportion
6
21.3 Graphical representation of direct
and inverse proportion
21.4 Compound measures
Key points
Internet Challenge 21
22Quadratic equations, curves and
inequalities
Starter: Solutions of equations
22.1 Solving quadratic equations –
factorising
22.2 Completing the square
22.3 Solving quadratic equations –
formula
22.4 Problems leading to quadratic
equations
22.5 Quadratic curves
22.6 Solving inequalities
Key points
Internet Challenge 22
23Advanced algebra
Starter: How many shapes?
23.1 Working with surds
23.2 Algebraic fractions
23.3 Cancelling common factors in
rational expressions
23.4 Simultaneous equations, one
linear and one quadratic
23.5 Changing the subject of an
equation where the symbol
occurs twice
23.6 Algebraic proofs
Key points
Internet Challenge 23
24Functions and function notation
Starter: Number crunchers
24.1 Introducing functions and
function notation
24.2 Domain and range
24.3 Inverse functions
24.4 Composite functions
Key points
Internet Challenge 24
Contents
25Further trigonometry
Starter: How tall is the church?
25.1 The sine rule
25.2 The cosine rule
1
25.3 Area of a triangle using }2}ab sin C,
and segments of circles
25.4 Trigonometry in 3-D
Key points
Internet Challenge 25
26Graphs and transformations
Starter: Making waves
26.1 Plotting and using graphs of curves
26.2 Graphs of sine, cosine and tangent
functions
Key points
Internet Challenge 26
27Vectors
Starter: Knight’s tours
27.1 Introducing vectors
27.2 Adding and subtracting vectors
27.3 Multiplying a vector by a number
(scalar multiplication)
27.4 Using vectors
Key points
Internet Challenge 27
28Calculus
Starter: Steeper and steeper
28.1 Gradient of a curve
28.2 Gradient of a curve –
differentiation
28.3 Harder differentiation
28.4 Maximum and minimum points
on curves
28.5 Further problems on maximum
and minimum
28.6 Distance, velocity and
acceleration
Key points
Internet Challenge 28
7
Chapter 9: Number sequences
CHAPTER 9
Number sequences
In this chapter you will learn how to:
• recognise and use common number sequences
• use rules to generate number sequences
• find a general formula for the nth term of an arithmetic sequence
Highlighted content is new
for this edition.
• find the sum of an arithmetic series.
You will also be challenged to:
Each chapter begins with a
STARTER: a problem-solving
exercise, activity or puzzle
designed to stimulate thinking
and discussion about some
of the ideas that underpin the
content of the chapter.
• investigate Fibonacci numbers.
Starter: Circles, lines and regions
Look at the sequence of circles below.
1
1
1
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
Pattern
Pattern
1Pattern
1 Pattern
1 1
1 point
1 point
1 point
1 point
0 lines
0 lines
0 lines
0 lines
1 region
1 region
1 region
1 region
2
1 2 12
2
2
2
2
Pattern
Pattern
2Pattern
2 Pattern
2 2
2 points
2 points
2 points
2 points
1 line1 line1 line1 line
2 regions
2 regions
2 regions
2 regions
1
1
2
1
2
1 1
3 3 3
2
1
2
2
1
Pattern
3Pattern
Pattern
3 Pattern
3 3
3 points
3 points
3 points
3 points
3 lines
3 lines
3 lines
3 lines
4 regions
4 regions
4 regions
4 regions
2
5
7
3
2
7 7
3 3 3
2 42 4 2 4 4
Pattern
Pattern
4Pattern
4 Pattern
4 4
4 points
4 points
4 points
4 points
6 lines
6 lines
6 lines
6 lines
8 regions
8 regions
8 regions
8 regions
The lines and regions are then counted. The lines and regions are not all the same
size.
Task 1
Describe a rule for how the number of points increases in this sequence.
Task 3
Describe a rule for how the number of regions increases.
8
4
7
The diagram shows a sequence of circles. Each circle has some points marked
around its circumference. Each point is joined to every other point by a line.
Task 2
Describe a rule for how the number of lines increases.
5
2 4 24
1 3 1 3 1 36 863 86 86 8
3
3 3 3
2 42 4 2 4 4
15 15
3
4
9.1 Number sequences
Task 4
Now draw pattern 5 and pattern 6, and see if your rules seem correct. You
should space out the points so that no triple intersections can occur, otherwise
you lose a region, for example:
No
Yes
9.1 Number sequences
Here are some number sequences that occur often in mathematics.
Name of sequence
First six terms
Formula for the nth term
Positive integers
1, 2, 3, 4, 5, 6, …
n
Even numbers
2, 4, 6, 8, 10, 12, …
2n
Odd numbers
1, 3, 5, 7, 9, 11, …
2n – 1
Square numbers
1, 4, 9, 16, 25, 36, …
n2
Cube numbers
1, 8, 27, 64, 125, 216, …
n3
Powers of 2
2, 4, 8, 16, 32, 64 …
2n
Powers of 10
10, 100, 1000, 10 000, 100 000,
1 000 000, …
10n
You may encounter these number patterns when solving mathematical problems
based on counting patterns.
EXAMPLE
Look at this pattern of squares.
Pattern 1
Pattern 2
Pattern 3
Pattern 4
a) How many squares would there be in pattern 5?
b) Find a formula for the number of squares in pattern n.
c) Use your formula to find the number of squares in pattern 100.
9
Chapter 9: Number sequences
SOLUTION
The number of squares forms a pattern 2, 4, 6, 8, that is, the even numbers.
a) Pattern 5 contains 2 3 5 5 10 squares.
b) Pattern n contains 2n squares.
c) Pattern 100 contains 2 3 100 5 200 squares.
Some number sequences are disguised versions of the common ones, perhaps
with a constant number added or multiplied.
EXAMPLE
Find the next three terms in this number sequence. Find also a formula for the
nth term.
101, 104, 109, 116, 125, …
SOLUTION
101, 104, 109, 116, 125, … are all 100 more than the square numbers.
The next three terms are 100 1 36, 100 1 49 and 100 1 64,
that is, 136, 149, 164
The nth term is 100 1 n2
EXERCISE 9.1
Write down the next two terms in each of these number sequences, and explain
how each term is worked out. Give an expression for the nth term in each case.
Questions to
test students’
understanding on each
sub-topic.
They are all related to the list of common sequences in the table on the previous page.
1 10, 20, 30, 40, 50, 60, …
2 5, 7, 9, 11, 13, 15, …
3 51, 53, 55, 57, 59, 61, …
4 4, 8, 12, 16, 20, 24, …
5 2, 8, 26, 80, 242, …
6 0.1, 0.01, 0.001, 0.0001, …
7 10, 30, 60, 100, 150, 210, …
8 2, 8, 18, 32, 50, 72, …
9 Look at this pattern of triangles.
Pattern 1
Pattern 2
Pattern 3
a) How many triangles would there be in pattern 7?
b) Find a formula for the number of triangles in pattern n.
10
Pattern 4
9.2 Describing number sequences with rules
10 Look at this pattern of spots.
Pattern 1
Pattern 2
Pattern 3
Pattern 4
a) Find an expression for the number of spots in pattern n.
b) How many spots would there be in pattern 30?
9.2 Describing number sequences with rules
It can be very useful to be able to describe number sequences using rules.
One way of doing this is to say how each term is connected to the next one
in the sequence. (This is sometimes called a term-to-term rule, because it
explains the link between one term and the next.)
Each sub-topic
within each chapter
is introduced, and
explained through
easy-to-follow worked
examples with
solutions.
EXAMPLE
A number sequence is defined as follows:
•
•
The first term is 3.
Each new term is double the previous one.
Use this rule to generate the first five terms of the number sequence.
SOLUTION
Start with 3:
33256
6 3 2 5 12
etc.
The first five terms of the sequence are 3, 6, 12, 24, 48, ….
EXAMPLE
A number sequence is defined as follows:
• The first term is 7.
• Each new term is 3 more than the previous one.
Use this rule to generate the first six terms of the number sequence.
11
Chapter 9: Number sequences
SOLUTION
Start with 7:
7 1 3 5 10
10 1 3 5 13
etc.
The first six terms of the sequence are 7, 10, 13, 16, 19, 22, ….
If you wanted to work out the 100th number in a sequence, it would be very
tedious to have to write out all 100 numbers, one at a time. In this case it
is better if you can use an algebraic expression for the nth term. (This is
sometimes called a position-to-term rule, since you can work out the value of
any term as long as you know its position in the sequence.)
EXAMPLE
The nth term of a number sequence is given by the expression 2n2 1 1.
a) Write down the first four terms of the sequence.
b) Find the value of the 20th term.
SOLUTION
a) n 5 1 gives 2 3 12 1 1 5 2 1 1 5 3
n 5 2 gives 2 3 22 1 1 5 8 1 1 5 9
n 5 3 gives 2 3 32 1 1 5 18 1 1 5 19
n 5 4 gives 2 3 42 1 1 5 32 1 1 5 33
The first four terms are 3, 9, 19, 33
b)When n 5 20, 2 3 202 1 1 5 800 1 1 5 801.
EXERCISE 9.2
1 A number sequence is defined as follows:
• The first term is 5.
• Each new term is 2 more than the previous one.
Use this rule to generate the first five terms of the number sequence.
2 A number sequence is defined as follows:
•
•
The first term is 1.
To find each new term, add 1 to the previous term, and double this total.
Use this rule to generate the first four terms of the number sequence.
3 The nth term of a number sequence is given by the expression 8n 2 1.
a) Write down the values of the first five terms.
b) Work out the value of the 20th term.
12
9.3 Arithmetic sequences
4 The nth term of a number sequence is given by the expression
a) Write down the values of the first six terms.
b) Work out the value of the 23rd term.
3n 1 1
.
2
5 Andy has been doing a mathematical investigation. He gets this sequence of numbers:
12, 15, 18, 21, 24, …
a) Describe Andy’s pattern in words.
b) Find the tenth term in Andy’s number sequence.
6 The nth term of a number sequence is given by the expression 100 2 n.
a) Write down the values of the first five terms.
b) Work out the value of the 50th term.
7 In a certain number sequence, the first term is 3. Each new term is found by multiplying the previous term
by 3.
a) Write down the first five terms of the number sequence.
b) What name is given to this particular number sequence?
8 The nth term of a number sequence is given by the formula 7n 1 3.
a) Work out the first three terms.
b) Find the value of the 10th term.
c) One of the numbers in the sequence is 1053. Which term is this?
9 The nth term of a number sequence is given by the expression n(n 1 1) .
2
a) Write down the values of the first four terms.
b) Work out the value of the 30th term.
c) Explain why all the terms in this sequence are integers.
d) What name is often given to the number sequence generated by this rule?
Problem-solving
question.
10 David is working with a number sequence. The nth term of his sequence is given by the expression 6n 1 7.
He gets the number 2770 as one of his terms. Show that David must have made a mistake.
9.3 Arithmetic sequences
A number sequence in which the terms go up or down in equal steps is called an arithmetic sequence.
The size of the step is called the common difference.
•
•
The first term of an arithmetic sequence is a.
The common difference is d.
When d is negative, each term is
EXAMPLE
less than the preceding term.
Callout boxes add
clarity: new for this
edition.
For each sequence, say whether it is arithmetic or not.
For each arithmetic sequence state the value of the first term, a, and the common difference, d.
a)
b)
c)
d)
2, 3, 5, 8, 12, …
2, 5, 8, 11, 14, …
1, 2, 4, 8, 16, …
40, 36, 32, 28, 24, …
Higher objective 3.1 A is covered
‘understand and use common difference (d)
and first term (a) in an arithmetic sequence’.
13
Chapter 9: Number sequences
SOLUTION
a) 2, 3, 5, 8, 12, … is not an arithmetic sequence.
Call-out boxes
add clarity: new for
this edition
The terms go up by 1, then 2,
then 3 and so on.
b) 2, 5, 8, 11, 14, … is an arithmetic sequence.
First term a 5 2 and common difference d 5 3
c) 1, 2, 4, 8, 16, … is not an arithmetic sequence.
d) 40, 36, 32, 28, 24, … is an arithmetic sequence.
The terms go up by 1, then 2,
then 4 and so on.
First term a 5 40 and common difference d 5 24
You can find the rule to give the nth term of an arithmetic sequence using the first term, a, and the
common difference, d.
Look at the arithmetic sequence with first term a 5 3 and common difference d 5 4.
Term number
Term
1
3
2
7
3
11
…
…
The first term is 3. To find The 3rd term is
each term, add 4 to the
3 1 2 lots of 4.
previous term.
20
79
…
…
The 20th term is
3 1 19 lots of 4.
n
3 1 (n 2 1) 3 4
The nth term is
3 1 (n 2 1) lots of 4.
Now look at the general arithmetic sequence with first term a and common difference d.
Term number
Term
1
a
2
a1d
3
a 1 2d
…
…
The first term is a. To
find each term, add d to
the previous term.
20
a 1 19d
The 20th term is
a 1 19 lots of d.
The nth term of an arithmetic sequence is
a 1 (n 2 1)d
where a is the first term and d is the common difference.
EXAMPLE
a) Find a formula for the nth term of the arithmetic sequence:
7, 10, 13, 16, 19, …
b) Find the 50th term of the sequence.
c) The nth term of the sequence is 205.
Find the value of n.
14
…
…
n
a 1 (n 2 1)d
The nth term is
a 1 (n 2 1) lots of d.
This material covers the new Edexcel
IGCSE Higher objective 3.1 B
‘know and use nth term = a + (n −1)d’
9.3 Arithmetic sequences
SOLUTION
a) a 5 7 and d 5 3
nth term is a 1 (n 2 1)d
nth term 5 7 1 (n 21)33
5 7 1 3(n 2 1)
5 7 1 3n 2 3
5 3n 1 4
b)Substitute n 5 50 into the formula for the nth term.
So the 50th term 5 33 50 1 4 5 154
c)3n 1 4 5 205
3n 5 201
n 5 67
So the 67th term is 205.
EXERCISE 9.3
1 The first five terms in an arithmetic sequence are:
12, 17, 22, 27, 32, …
a) Find the value of the 10th term.
b) Write down, in terms of n, an expression for the nth term of this sequence.
2 The first four terms in an arithmetic sequence are:
58, 50, 42, 34,…
a) Find the value of the first negative term.
b) Write down, in terms of n, an expression for the nth term of this sequence.
Here are some arithmetic sequences. For each one, find, in terms of n, an expression for the nth term of the
sequence.
3 8, 11, 14, 17, 20, …
4 2, 7, 12, 17, 22, …
5 10, 9, 8, 7, 6, …
6 4, 9, 14, 19, 24, …
7 21, 24, 27, 30, 33, …
8 12, 10, 8, 6, 4, …
9 Nina has been making patterns with sticks. Here are her first three patterns.
Pattern 1
4 sticks
Pattern 2
7 sticks
Pattern 3
10 sticks
a) Work out the number of sticks in pattern 6.
b) Write down, in terms of n, an expression for the nth term of this sequence.
c) Explain how the coefficients in your formula are related to the way the sticks fit together.
15
Chapter 9: Number sequences
10 The tenth term of an arithmetic sequence is 68 and the eleventh term is 75.
a) Write down value of the common difference for this sequence.
b) Work out the value of the first term.
c) Write down, in terms of n, an expression for the nth term of this sequence.
Check that your formula works when n 5 10 and n 5 11.
9.4 Arithmetic series
When you add terms of a sequence it is called a series.
So 1, 4, 7, …
is an arithmetic sequence.
1 1 4 1 7 1 … is an arithmetic series.
There is a neat method you can use to add the terms in an arithmetic series.
Look at this method of adding all the numbers from 1 to 100.
Write the
series out
forwards
…
… then
backwards.
1
1
2
1
3
1
4 …
1
98
1
99
1
100
100
1
99
1
98
1
97 …
1
3
1
2
1
1
101
1 101
1 101
1
101 …
1 101
1
101
1 101
You now have 100
terms which are all
the same!
Add together
the sequences
The sum of both series is 100 3 101 5 10 100
So the sum of the numbers from 1 to 100 is ½ 3 10 100 5 5050
You can use the same method to find the sum of the terms, Sn, of any arithmetic series with first term a and
common difference d.
Write the series out
forwards …
Sn
5 a
Sn
5 (a 1 (n 2 1) d) 1
2 3 Sn 5 2a 1 (n 21 )d
1 (a 1 d)
1 (a 1 2d)
1 … 1 (a 1 (n 2 1) d)
(a 1 (n 2 2) d) 1 (a 1 (n 2 3) d ) 1 … 1
1 2a 1 (n 21) d
1 2a 1 (n 21) d
1 …
a
… then backwards.
1 2a 1 (n 2 1) d
So you have n terms which are all the same; the sum of these terms is:
2 3 Sn 5 n 3 (2a 1 (n 2 1)d
This is the sum of two identical series, so you need to halve it:
n
(2a 1 (n 2 1)d )
Sn 5
You will be given this
2
formula in the exam.
16
This material covers the
new Edexcel IGCSE Higher
objective 2.2 E
‘use algebra to support and
construct proofs’.
9.4 Arithmetic series
EXAMPLE
The first term of an arithmetic series is 40 and the common difference is 2½.
Find the sum of the first 21 terms of the arithmetic series.
SOLUTION
n
Substitute a  40 , d  2½ and n  21 into Sn  2 (2a 1 (n 2 1)d )
S21 
1
21
3 (2 3 40 1 (21 2 1) 3 (2 2 ))
2
(
1
)
 10.5 3 (80 1 20 3 2 2 )
 10.5 3 (80 2 10)
 735
Some problems are more complicated – be prepared to solve simultaneous equations when tackling
questions on arithmetic sequences and series.
EXAMPLE
The 6th term of an arithmetic series is 20.
The 11th term of the same arithmetic series is 35.
Find the sum of the first 100 terms of this arithmetic series.
SOLUTION
The nth term of an arithmetic series is a 1 (n 2 1)d.
The 6th term is 20, so a 1 5d  20
and the 11th term is 35, so a 1 10d  35
You now have two equations and two unknowns, so you can solve them simultaneously to find a and d.
a 1 10d  35
− a 1 5d  20
5d  15
Now substitute d  3 into
Subtract to eliminate a.
so d  3
a 1 5d  20
a 1 5 3 3  20
so a  5
You can use the formula for the sum of a series, substitute a  5, d  3 and n  100 into
n
Sn  2 (2a 1 (n 2 1)d )
100
3 (2 3 5 1 (100 2 1) 3 3)
2
 50 3 (10 1 99 3 3)
S100 
 50 3 307
 15350
17
Chapter 9: Number sequences
EXERCISE 9.4
New Exercise to cover new
specification objectives.
1 Find the sum of the first 20 terms of each of the following series.
a) first term is 5, common difference is 3
b) first term is 3, common difference is 5
c) first term is 3, common difference is 25
d) first term is 5, common difference is 23
2 The first three terms in an arithmetic sequence are 7, 9, 11.
Find
a) an expression for the nth term of the sequence
b) the 20th term
c) the sum of the first 50 terms.
3 Find the sum of the whole numbers from 1 to 1000.
4 Find the sum of the first 100 multiples of 3.
5 The first term in an arithmetic series is 3.
The sum of the first three terms is 21.
a) Find the common difference.
The last term is 99.
b) How many terms are in the series?
c) Find the sum of all the terms of the series.
Questions 5–8 involve
problem solving.
6 The 12th term of an arithmetic series is 62.
The 20th term of the series is 102.
Find the sum of the first 20 terms of this arithmetic series.
7 The 6th term of an arithmetic series is 20.
The 11th term of the series is 35.
Find the sum of the first 100 terms of this arithmetic series.
8 The second term of an arithmetic series is 95 and the fourth term is 91.
a) Find the first term and the common difference.
b) Find the sum of the first 100 terms.
c) Sn is the sum of the first n terms.
What is the maximum value of Sn?
REVIEW EXERCISE 9
Review exercises test
understanding for the
chapter as a whole.
Find the next three terms in each of these number sequences. For those
that form arithmetic sequences, write down, in terms of n, an expression
for the nth term of this sequence.
1 11, 22, 33, 44, 55, …
2 2, 4, 8, 16, 32, …
3 2, 5, 8, 11, …
4 1, 4, 9, 16, 25, …
5 10, 9, 8, 7, 6, …
6 100, 99, 97, 94, 90, …
7 A number sequence is defined as follows:
• The first term is 7.
• To get each new term, multiply the previous one by 3 and subtract 15.
Work out the first four terms of this sequence.
18
Review exercise 9
8 The nth term of a number sequence is given by the expression
a) Work out the first five terms of this sequence.
b) Do the first five terms form an arithmetic sequence?
n2 1 3n .
2
9 Timothy has been drawing patterns. Here are his first three patterns.
Pattern 1
6 sticks
Pattern 2
11 sticks
Pattern 3
16 sticks
a) Write down the number of sticks in pattern 5.
b) Work out the number of sticks in pattern 12.
c) Write down, in terms of n, an expression for the nth term of this sequence.
10 Find i) the nth term and ii) the sum of the first 20 terms of each of the following arithmetic series.
a)
b)
c)
d)
first term is 9, common difference is 4
first term is 4, common difference is 9
first term is 90, common difference is –4
first term is 50, common difference is –9
11 The first three terms in an arithmetic sequence are 15, 19, 24.
Find
a) an expression for the nth term of the sequence
b) the 20th term
c) the sum of the first 50 terms.
New questions for this
edition.
12 Find the sum of the first 50 multiples of 5.
13 The first term in an arithmetic series is 3.
The sum of the fifth term and the sixth term is 42.
a) Find the common difference.
The last term is 499.
b) How many terms are in the series?
c) Find the sum of all the terms of the series.
14 The 11th term of an arithmetic series is 55.
The 21st term of the series is 50.
Find the first term and the sum of the first 100 terms of this arithmetic series.
15 Here are the first five terms of a sequence.
30, 29, 27, 24, 20, …
a) Write down the next two terms in the sequence.
Here are the first five terms of a different sequence.
1, 5, 9, 13, 17, …
b) Find, in terms of n, an expression for the nth term of the sequence.
Past-paper questions
are indicated.
[Edexcel]
19
Chapter 9: Number sequences
16 Here are the first five numbers of a simple sequence.
1, 5, 9, 13, 17
a) Write down the next two numbers of the sequence.
b) Write down, in terms of n, an expression for the nth term of this sequence.
[Edexcel]
17 Here are the first five terms of an arithmetic sequence.
6, 11, 16, 21, 26
[Edexcel]
Find an expression, in terms of n, for the nth term of this sequence.
A summary of essential
points from the chapter.
Key points
1 Common number sequences include the positive integers, the even numbers and
the odd numbers. Others you should learn to recognise are:
Square numbers
Cube numbers
Powers of 2
Powers of 10
Triangular numbers
1, 4, 9, 16, 25, 36, …
1, 8, 27, 64, 125, 216, …
1, 2, 4, 8, 16, 32, …
1, 10, 100, 1000, 10 000, 100 000, …
1, 3, 6, 10, 15, 21, …
2 A term-to-term rule is a rule which explains how to find the next term in a
sequence using the term before.
3 A position-to-term rule is a rule that gives an expression for the nth term of a
sequence.
4 In an arithmetic sequence there is a constant difference between successive
terms.
5 The nth term of an arithmetic sequence is
a + (n − 1)d
where a is the first term and d is the common difference.
6 A series is the sum of a sequence.
5, 7, 9, … is an arithmetic sequence.
5+7+9…
is an arithmetic series.
7 The sum Sn of the first n terms of an arithmetic series is
Sn = n (2a + (n − 1)d)
2
where a is the first term and d is the common difference.
20
Internet Challenge 9
Internet Challenge 9
Fibonacci numbers
Fibonacci numbers are used to model the behaviour of living systems. Fibonacci numbers also lead to
the Golden Ratio, widely used in classical art and architecture. In this challenge you will need to use a
spreadsheet at first, before looking on the internet to complete your work.
Here is the Fibonacci number sequence:
1, 1, 2, 3, 5, 8, 13, 21, ….
1 Type these numbers into a computer spreadsheet, such as Excel. (It is a good idea to enter them in a
vertical list, rather than a horizontal one.)
2 Each term (apart from the first two) is found by adding together the two previous ones, for example,
13 5 8 1 5. Use your spreadsheet replicating functions to automatically generate a list of the first
50 Fibonacci numbers.
3 Divide each Fibonacci number by the one before it, for example 8 4 5 5 1.6. Set up a column on your
spreadsheet to do this up to the 50th Fibonacci number. What do you notice?
The quantities you found in question 3 approach a limit called the Golden Ratio, f.
4 Using your spreadsheet value for f, calculate 1 2 f and
1
f .What do you notice?
Now use the internet to help answer the following questions. Find pictures where appropriate.
5 How was the Golden Ratio used by the builders of the Parthenon in Athens?
6 Whose painting of ‘The Last Supper’ was based on Golden Ratio constructions?
Links to
mathematics
outside of the
classroom.
7 Which painter was said to have ‘attacked every canvas by the golden section’?
8 When was Fibonacci born? When did he die?
9 Is there a position-to-term rule for Fibonacci numbers, that is, is there a formula for finding the nth
number?
10 What sea creature has a spiral shell that is often (mistakenly) said to be based on a Golden Ratio spiral?
Each chapter concludes with an Internet Challenge, for students to work
through either at school or at home.
This problem solving or research activity frequently goes beyond the strict
boundaries of the IGCSE specification, providing enrichment and leading to
a deeper understanding of mainstream topics.
The Challenges may look at the history of mathematics and mathematicians, or
the role of mathematics in the real world.
When doing these, it is hoped that students will not just answer the
written questions, but also take the time to explore the subject a little deeper.
21
Chapter 9: Number sequences
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9.3 Arithmetic sequences
International GCSE (9-1) Mathematics
Practice for Edexcel Third Edition
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INTERNATIONAL
Chapter 9: Number sequences
GCSE
(9–1)
Mathematics
for Edexcel Specification A
This sample chapter is taken from Edexcel
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Student Book Third Edition, which has
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