DEPARTMENT OF MATHEMATICS
Core 1
Topic 8 – Sequences and Series
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Sequences and Series
Sigma Notation
Recurrence Relations
Write in full and simplify a
sequence given in sigma notation
(Σ)
Arithmetic Series
Find the next term in a recurrence
relation given the previous term
Work backwards with a recurrence
relation to find the previous term
Find the value of a constant in a
recurrence relation - sigma
notation and substitution
Functional Arithmetic Series
Identify the first term, a,
and the common difference, d, in
an arithmetic series
Find the nth term of an arithmetic
series using the formula:
𝑈𝑛 = 𝑎 + 𝑛 − 1 𝑑
Recognise and solve an nth term
problem
Recognise and solve a Sum problem
(Including 'fixed earnings'
questions)
Find the Sum of an arithmetic
series using the formula:
𝑛
𝑆𝑛 = [2𝑎 + 𝑛 − 1 𝑑]
2
Know when to use the alternative
Sum formula:
𝑛
𝑆𝑛 = [𝑎 + 𝐿]
2
Solve simultaneous equations
involving nth terms and sums
Write an equation involving a and d
given specific terms
Find a sum from Sigma notation
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Sequences and Arithmetic Series
We’re onto topic 8 now – Sequences and Arithmetic Series. Firstly, we recap Sigma
notation, which we covered in the introduction lesson this year.
Sigma Notation:
Sigma (written here in its capital letter form, Σ) represents ‘sum of’. We use it to
write out a long sum in a quick and easy manner:
Sigma Notation:
What does everything mean?
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∑(𝑛2 − 2𝑛)
𝑛=1
Worked Examples:
Evaluate:
3
𝑎)
∑(4𝑛2 + 1)
𝑛=0
3
4
𝑏)
)𝑛
∑ (−1
𝑛=1
6𝑛
(
)
𝑛+1
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Notes:
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Recurrence Relations:
When you know the rule to get from one term to the next, you can use this
information to produce a recurrence relationship.
In the exam, we’ll be given the recurrence relation; in short, they’ll tell us how
we get to the next number in the sequence.
Here, we’ll see some new notation. 𝑈𝑛 represents the ‘nth term’ in our sequence –
i.e. 𝑈1 represents the 1st term, 𝑈2 represents the 2nd term, 𝑈𝑛+1 would represent
the n+1th term (the term after the nth term).
Worked Examples:
a)
A sequence is defined by the recurrence relation 𝑈𝑛+1 = 𝑈𝑛 + 4, and 𝑈1 = 7.
Find 𝑈2 and 𝑈3
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b)
A sequence is defined by the recurrence relation 𝑈𝑛+1 = 2𝑈𝑛 − 3, and 𝑈1 = 5.
Find 𝑈2 and 𝑈3
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c)
A sequence is defined by the recurrence relation 𝑈𝑛+1 = 5𝑈𝑛 + 1.
Given that 𝑈2 = 11, what was the value of 𝑈1 ?
d)
A sequence is defined by the recurrence relation 𝑈𝑛+2 = 3𝑈𝑛+1 − 𝑈𝑛 ,
where 𝑈1 = 4 and 𝑈2 = 2
What are the values of 𝑈3 and 𝑈4 ?
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Now, let’s have a peek at 2 tougher examples, taken directly from past exams:
Worked Examples:
1)
[Jan 2012]
A sequence, 𝑥1 , 𝑥2 , 𝑥3 , … is defined by:
𝑥1 = 1
and
𝑥𝑛+1 = 𝑎𝑥𝑛 + 5, 𝑛 ≥ 1
where 𝑎 is a constant.
a)
Write down an expression for 𝑥2 in terms of 𝑎
b)
Show that 𝑥3 = 𝑎2 + 5𝑎 + 5
c)
Given that 𝑥3 = 41, find the possible values of 𝑎
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2)
[Jun 2014]
A sequence of numbers, 𝑎1 , 𝑎2 , 𝑎3 , … is defined by: 𝑎𝑛+1 = 5𝑎𝑛 − 3,
𝑛≥1
Given that 𝑎2 = 7
a)
Find the value of 𝑎1
b)
Find the value of ∑4𝑟=1 𝑎𝑟
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Notes:
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Arithmetic Series:
A sequence that increases by a constant amount each time is called an arithmetic
sequence.
Examples include:
3, 7, 11, 15, …
2, 7, 12, 17, 22, …
17, 14, 11, 8, …
𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑, …
The sequence 2, 4, 8, 16, … is not arithmetic – it goes up by a multiple of 2 each
time, but not by a constant amount (it’s actually a geometric sequence, which
we’ll study in C2).
 The first term of an arithmetic series is represented by the letter 𝒂
 The common difference between each term in the series (i.e. the ‘constant
amount’) is represented by the letter 𝒅
Therefore, all arithmetic series can be written in the following manner –
𝑎 + (𝑎 + 𝑑) + (𝑎 + 2𝑑) + (𝑎 + 3𝑑) + (𝑎 + 4𝑑) + (𝑎 + 5𝑑) + …
This gives us the formula for finding the nth term of an arithmetic series:
𝑈𝑛 = 𝑎 + (𝑛 − 1)𝑑
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Worked Examples:
Using the nth term formula for an arithmetic series:
1)
The series is:
4 + 7 + 10 + 13 + ⋯
a)
What is the 20th term?
b)
What is the 50th term?
2)
The series is:
a)
What is the 10th term?
b)
What is the 40th term?
100 + 93 + 86 + 79 + ⋯
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3)
The arithmetic series is given by: 5 + 9 + 13 + 17 + 21 + ⋯ + 805
a)
How many terms are there?
b)
Which term of the series would be 129?
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Notes:
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Solving Simultaneous Equations involving the nth term formulae:
We’ll often be given 2 pieces of information and expected to form 2 simultaneous
equations from this. After this, we’ll solve the simultaneous equations to find out
the values of 𝑎 and 𝑑:
Worked Example:
1)
Given that the 3rd term of an arithmetic series is 20 and the 7th term is 12:
a)
Find the first term, 𝑎
b)
Find the 20th term
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The Sum of an Arithmetic Series:
We also need to be able to find the sum of an arithmetic series.
There is a quick way to find the sum of an arithmetic series.
Carl Friedrich Gauss is given credit for this method – whilst at
junior school, legend has it that a cruel teacher would waste
time within lessons by asking the students to add up ridiculous
sums, such as 1 + 2 + 3 + 4 + 5 + … + 100.
Gauss spotted a quick way to do this, and we’ll look at this.
We’ll also derive the formula for any arithmetic series.
(This is a proof that you are expected to know for the exam.)
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Notes:
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The really helpful part of this is that both formulae are included on the formula
sheet for C1. We aren’t expected to remember them, but we are expected to
recognise when and how to apply them.
Note that there is a 3rd formula that we haven’t discussed yet – we’ll look at this a
little later on.
Let’s consider some examples where we are required to apply the sum formula:
Worked Examples:
1)
Find the sum of the first 100 odd numbers
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2)
An arithmetic series is given as:
4 + 9 + 14 + 19 + ⋯
How many terms does it require so that the sum exceeds 2000?
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Of course, one to be careful with – read the Q carefully and don’t fall into a trap:
3)
Robert starts a new job with a salary of £15000. He is promised a pay rise
of £1000 per year, up to a maximum salary of £25000.
How much will he earn after:
a)
b)
8 years
14 years?
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Notes:
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Finding a sum from a Sigma Notation:
We can use the sum formula to quickly solve large sigma notation sums.
Worked Example:
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𝑎)
∑(4𝑟 + 1)
𝑟=1
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Functional Arithmetic Series:
This style of question applies the formulae seen within this topic in a real-life
context. We’ll also have an opportunity to see how we could use the additional
sum formula from the formula sheet that we haven’t used thus far.
Worked Example:
[Jan 2013]
Lewis played a game of space invaders. He scored points for each spaceship that
he captured.
Lewis scored 140 points for capturing his first spaceship.
He scored 160 points for capturing his second spaceship, 180 points for capturing
his third spaceship, and so on.
The number of points scored for capturing each successive spaceship formed an
arithmetic sequence.
(a)
Find the number of points that Lewis scored for capturing his 20th
spaceship.
(b)
Find the total number of points Lewis scored for capturing his first 20
spaceships.
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Sian played an adventure game. She scored points for each dragon that she
captured. The number of points that Sian scored for capturing each successive
dragon formed an arithmetic sequence.
Sian captured n dragons and the total number of points that she scored for
capturing all n dragons was 8500.
Given that Sian scored 300 points for capturing her first dragon and then 700
points for capturing her nth dragon,
(c) find the value of n.
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Notes:
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Sequences and Arithmetic Series – Questions
Exercise 1:
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Exercise 2:
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Exercise 3:
Exercise 4:
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Exercise 5:
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Exercise 6:
Exercise 7:
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See additional sheets for past exam questions.
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Solutions to all exercises:
Exercise 1:
Exercise 2:
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Exercise 3:
Exercise 4:
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Exercise 5:
Exercise 6:
Exercise 7:
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