A sequence is a list of terms. For example, 3, 6, 9, 12, 15, …
A series is the sum of a list of terms. For example, 3 + 6 + 9 + 12 + 15 + …
The terms of a sequence are separated by a comma, while with a series they are all added together.
Here are some important definitions prefacing the content in this chapter:
▪A sequence is increasing if each term is greater than the previous.
e.g. 4, 9, 14, 19, …
▪A sequence is decreasing if each term is less than the previous.
e.g. 5, 4, 3, 2, 1, …
▪A sequence is periodic if the terms repeat in a cycle; 𝑢𝑛+𝑘 = 𝑢𝑛 for some k, which is known as the
order of the sequence. e.g. -3, 1, -3, 1, -3, … is periodic with order 2.
An arithmetic sequence is one where there is a common difference between each term. Arithmetic sequences
are of the form
𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑, …
where 𝑎 is the first term and 𝑑 is the common difference.
•The nth term of an arithmetic series is given by: 𝒖𝒏= 𝒂 + (𝒏−𝟏)𝒅
An arithmetic series is the sum of the terms of an arithmetic sequence.
•The sum of the first 𝑛 terms of an arithmetic series is given by 𝑺𝒏=𝒏
𝟐[𝟐𝒂+(𝒏 − 𝟏)𝒅] or
𝑺𝒏=𝒏
𝟐(𝒂+ 𝒍)
where 𝑎 is the first term, 𝑑 is the common difference and 𝑙 is the last term.
You need to be able to prove this result. Here is the proof:
The defining feature of a geometric sequence is that you must multiply by a common ratio, r, to get from
one term to the next. Geometric sequences are of the form
𝑎, 𝑎𝑟, 𝑎𝑟2, 𝑎𝑟3, 𝑎𝑟4, …
where 𝑎 is the first term in the sequence and 𝑟 is the common ratio.
•The nth term of a geometric sequence is given by: 𝒖𝒏= 𝒂𝒓𝒏−𝟏
It can help in many questions to use the fact that 𝑢𝑘+1
𝑢𝑘=𝑢𝑘+2
𝑢𝑘+1 = 𝑟. This is especially helpful when the
terms of the sequence are given in terms of an unknown constant. Part a of example 4 highlights this.
A geometric series is the sum of the terms of a geometric sequence.
•The sum of the first n terms of a geometric series is given by:
𝑺𝒏=𝒂(𝟏− 𝒓𝒏)
𝟏− 𝒓
by multiplying the top and bottom of the fraction by -1, we can also use
𝑺𝒏=𝒂(𝒓𝒏−𝟏)
𝒓− 𝟏
You need to be able to prove this result. Here is the proof:
Since division by zero is undefined, this formula is invalid when 𝑟 = 1.
The sum to infinity of a geometric sequence is the sum of the first n terms as n approaches infinity. This does not
exist for all geometric sequences. Let’s look at two examples:
2 + 4 + 8+ 16 +32 + ⋯
Each term is twice the previous (i.e. 𝑟 = 2). The sum of such a series is not finite, since each term is bigger than
the previous. This is known as a divergent sequence.
2 + 1 + 1
2+1
4+1
8+⋯
Here, each term is half the previous (i.e. 𝑟 = 1
2). The sum of such a series is finite, since as n becomes large, the
terms will tend to 0. This is known as a convergent sequence.
▪A geometric sequence is convergent if and only if |𝑟| < 1.
The sum to infinity of a geometric sequence only exists for convergent sequences, and is given by:
𝑺∞=𝒂
𝟏−𝒓
A recurrence relation is simply another way of defining a sequence. With recurrence relations, each term is given as a function of
the previous. For example, 𝑢𝑛+1 = 𝑢𝑛+ 4, 𝑢1= 1 represents an arithmetic sequence with first term 1 and common difference
4. In order to generate a recurrence relation, you need to know the first term.
You need to be comfortable solving problems where series are given in sigma notation. Below is an annotated example explaining
how the sigma notation is used.
∑(7− 2𝑟)= 5 +3 +1 −1 +⋯
20
𝑟=1
If you are ever troubled by a series given in sigma notation, it is a good idea to write out the first few terms and analyse the series
that way.
Geometric and arithmetic sequences are often used to model real-life scenarios. Consider the amount of money in a savings
account; this can be modelled by a geometric sequence where r represents the interest paid at the end of each year and a is the
amount of money in the account at the time of opening.
You need to be able to apply your knowledge of sequences and series to questions involving real-life scenarios. It is important to
properly understand the context given to you, so take some time to read through the question more than once.
Sequences and Series Cheat Sheet
This tells us the last value of r for our
sequence, i.e. our last term will be
7−2(20)= −33
This is the value of r where our
series starts, i.e. our first term is
7 −2 (1) = 5.
This series in particular represents an
arithmetic series with first term 5
and common difference -2
Inputting 𝑟 = 1 into this expression gives us the first
term, 𝑟 = 2 gives us the second term, and so forth.
Factorising out 𝑆𝑛 from the LHS
and 𝑎 from the RHS
Pure Year 2
Example 1: Prove that the sum of the first n terms of an arithmetic series is 𝑆𝑛=𝑛
2[2𝑎 + (𝑛− 1)𝑑].
We start by writing the sum out normally [1], and then in reverse [2]:
[1]𝑆𝑛= 𝑎 +(𝑎 + 𝑑)+(𝑎 + 2𝑑)+ ⋯ + (𝑎 + (𝑛− 2)𝑑)+(𝑎 + (𝑛 − 1)𝑑)
[2]𝑆𝑛=(𝑎 +(𝑛 − 1)𝑑)+(𝑎 + (𝑛 − 2)𝑑)+⋯+(𝑎 + 2𝑑)+(𝑎 + 𝑑)+𝑎
Adding [1] and [2] gives us:
[1]+[2]: 2𝑆𝑛= 𝑛(2𝑎 + (𝑛 − 1)𝑑)
∴ 𝑆𝑛=𝑛
2[2𝑎 + (𝑛− 1)𝑑]
Example 2: The fifth term of an arithmetic series is 33. The tenth term is 68. The sum of the first n
terms is 2225. Show that
7𝑛2+ 3𝑛 −4450 = 0
, and hence find the value of n.
5𝑡ℎ 𝑡𝑒𝑟𝑚 𝑖𝑠 33 ∴ 𝑎 + 4𝑑 = 33 [1] (nth term formula)
10𝑡ℎ 𝑡𝑒𝑟𝑚 𝑖𝑠 68 ∴68 = 𝑎 + 9𝑑 [2] (nth term formula)
Solving [1] and [2] simultaneously, we find that 𝑑 = 7 and 𝑎 = 5.
Sum of first n terms is 2225 ∴ 𝑛
2[2𝑎 + (𝑛− 1)𝑑]=2225
𝑛
2[10 +(𝑛−1)(7)]= 2225
𝑛(3+7𝑛)= 2225(2)
7𝑛2+ 3𝑛 −4450 = 0
To find the value of n, we just need to solve the quadratic. Using the quadratic formula, we find that
𝑛 = 25 𝑜𝑟 𝑛 = −25.4
. Since the term number must be a positive integer, we can conclude that
𝑛 =
25.
Example 3: Prove that the sum of the first n terms of a geometric series is
𝑆𝑛=𝑎(1−𝑟𝑛)
1−𝑟
𝑆
𝑛
= 𝑎 + 𝑎𝑟 + 𝑎𝑟
2
+⋯+𝑎𝑟
𝑛−1 [1]
multiplying the sum by 𝑟 𝑟𝑆𝑛=𝑎𝑟 +𝑎𝑟2+ 𝑎𝑟3+ ⋯ + 𝑎𝑟𝑛 [2]
Subtracting [2] from [1] 𝑆𝑛−𝑟𝑆𝑛= 𝑎 − 𝑎𝑟𝑛
⇒ 𝑆
𝑛
(1−𝑟)= 𝑎(1 − 𝑟
𝑛
)
Factoring out
𝑆
𝑛 and
𝑎
∴ 𝑆𝑛=
𝑎(1−𝑟𝑛)
1−𝑟
Dividing by
1−𝑟
𝑟 = 1
Example 4: The first three terms of a geometric series are (k-6), k, (2k+5), where k is a positive constant.
a) Show that
𝑘2− 7𝑘 −30 = 0
b) Hence find the value of k.
c) Find the common ratio of this series and hence calculate the sum of the first 10 terms.
d) Find the sum to infinity for a series with first term k and common ratio 𝑘−6
𝑘.
a) Using the fact that 𝑢𝑘+1
𝑢𝑘=𝑢𝑘+2
𝑢𝑘+1 = 𝑟
𝑘
𝑘 − 6 =2𝑘 + 5
𝑘
Cross-multiplying and simplifying:
⇒ 𝑘2=(2𝑘 + 5)(𝑘 − 6)
⇒ 𝑘2= 2𝑘2− 7𝑘 − 30
⇒ 𝑘2− 7𝑘 − 30 = 0 as required.
b) Solving the quadratic:
(𝑘 − 10)(𝑘 + 3)= 0 ⇒ 𝑘 = 10, 𝑘 = −3
Since we are told k is positive, we can conclude 𝑘 = 10.
c) From part a, 𝑢𝑘+1
𝑢𝑘= 𝑟 = 10
10−6 =5
2
𝑆10 = 𝑎(1−𝑟10)
1−𝑟 =4(1−(2.5)10)
1−2.5 =25428.6 ⇒ 25400 𝑡𝑜 3 𝑠.𝑓 .
d) 𝑎 = 10 𝑎𝑛𝑑 𝑟 = 1
5
2=2
5
∴ 𝑆∞=10
1−2
5=50
3
Example 5: The sequence with recurrence relation 𝑢𝑘+1 = 𝑝𝑢𝑘+𝑞, 𝑢1= 5, where 𝑝 is a constant and 𝑞 = 13, is periodic with
order 2. Find the value of
𝑝
.
We know the order is 2. So if 𝑢1= 5, then 𝑢3= 5
too. Finding 𝑢3:
𝑢2= 𝑝𝑢1+13 = 5𝑝 + 13
𝑢3= 𝑝𝑢2+13 = 𝑝(5𝑝 + 13)+13
Equating to 5:
𝑝(5𝑝 + 13)+13 = 5
Simplifying:
5𝑝2+13𝑝 + 8 = 0
Solving the quadratic by factorising:
We get 2 values, one of which is correct.
(5𝑝 + 8)(𝑝 +1)= 0
𝑝 = −1 or 𝑝 = −1.6
Substitute 𝑝 = −1.6 and 𝑝 = −1 into the
recurrence relation separately to see which one
correctly corresponds to a periodic sequence of
order 2.
Substituting 𝑝 = −1.6 into the recurrence relation gives a
sequence where each term is 5 and so does not have order 2.
Using
𝑝 = −1
does give us a periodic sequence with order 2
however, so 𝑝 = −1.
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Ex
ample 6: A virus is spreading such that the number of people infected increases by 4% each day. Initially 100 people were
d
ia
gnosed with the virus. How many days will it be before 1000 are infected?
𝑎 = 100 and 𝑟 = 1.04. We are really just trying to find the smallest value of n such
that
U
𝑛 > 1000.
1. divide both side by 100
2. take logs of both sides
3. divide both side by log(1.04) to solve for n (note that log(10) = 1)
4. round your answer up