Sequences And Series
Introduction
After reading this article on sequences and series you will be able to:
- To understand the structure of a sequence
- To be able to recognise the different types of series
- To be able to apply the formula for arithmetic sequences
Sequences and series - What is a sequence?
A sequence is a set of numbers in any particular order.
For instance 1, 2, 3, 4, 5…. is a sequence.
Each number in the sequence is known as a term.
We can the terms of a sequence in terms of algebra in this is written as a_1, a_{2,} a_3, \ldots a_k
So for the sequence that we have mentioned: a_{1,}=1, a_{2,}=2 \mathrm{etc}
When the terms of the sequence are added together as follows: 1 + 2 + 3 + 4 + 5 + ….., then the resulting sum is known as a series.
Quite often the letter sigma is used to represent summation or the adding up of terms of a sequence.
So this means that a_1+a_2+a_3+a_4+a_5 can be written as \sum_{i=1}^{k=5} a_k or just \sum_{i=1}^5 a_k
Sequences and series - Types of sequences
- A sequence which can increase or decrease by a fixed amount either by addition or subtraction is known as an Arithmetic Sequence. The fixed amount is known as the common difference.
Consider the following sequence: 8 11 14 17
You can see here that to obtain the next term you simply need to add three to the previous term.
- A sequence which can increase or decrease by a fixed amount by multiplication only is known as a Geometric Sequence. The fixed amount is known as the common ratio.
Consider the sequence: 2 4 8 16
You will see here that to obtain the next term you simply need to multiply the previous term by 2.
However if we consider the sequence: 16 8 4 2 here you need to multiply the previous term by \frac{1}{2}
Remember to obtain the next term in a geometric sequence is done through multiplication only.
- A sequence which repeats itself at regular intervals is known as a periodic sequence.
Consider the sequence 1, 2, 3, 1, 2, 3, 1, 2, 3…..
You will see here that the sequence is repeating itself and the pattern repeats itself after every three terms. We can see that the period of this sequence is 3.
- A sequence which lies above or below a certain value is known as a oscillating sequence.
Consider the sequence 5, 6, 5, 4, 5, 6, 5, 4, ….
You will see here that this sequence is above and below the value 5. So the sequence is oscillating about 5.
Example
A sequence is defined as a_k=2+3 k \text { for } k=1,2,3 \ldots
Find the first three terms.
Solution
This is a case of substitution \begin{aligned} & k=1: a_1=2+3(1)=5 \\ & k=2: a_2=2+3(2)=8 \\ & k=3: a_3=2+3(3)=11 \end{aligned}
Sequences and series – Example
Find the first 4 terms and state the type of sequence that this is.
What is \sum_{k=1}^4 a_k ?
Solution
Substituting k=1,2,3,4 into the general expression gives us:
\begin{gathered} a_1=3-1=2 \\ a_2=6-1=5 \\ a_3=9-1=8 \\ a_4=12-1=11 \end{gathered}You can see that the sequence is increasing by the fixed amount of three each time so we have an arithmetic sequence.
\sum_{k=1}^4 a_k=a_1+a_2+a_3+a_4=2+5+8+11=26Example
A sequence is defined by a_k=(-1)^k \text { for } k=1,2,3 \ldots
Write down the first four terms and state the type of sequence. What is \sum_{k=1}^4 a_k?
Solution
Substituting k=1, 2, 3, 4 into the general expression gives us:
\begin{gathered} a_1=(-1)^1=-1 \\ a_2=(-1)^2=1 \\ a_3=(-1)^3=-1 \\ a_4=(-1)^4=1 \end{gathered}You will see here that the sequence is geometric with a common ratio of -1 and it is also oscillating with a period of 2.
\sum_{k=1}^4 a_k=a_1+a_2+a_3+a_4=-1+1-1+1=0Sequences and series – example:
Example
A sequence is defined by a_{k+1}=3 a_k where a_1=1
What are the first four terms of the sequence?
Solution
With this type of sequence in order to obtain the next term the previous term must be known.
You are given the first term i.e a_1=1
And to find the next terms we substitute k=1, 2, 3….
\begin{aligned} & k=1: a_2=3 a_1=3(1)=3 \\ & k=2: a_3=3 a_2=3(3)=9 \\ & k=3: a_4=3 a_3=3(9)=27 \end{aligned}So the first four terms are: 1, 3, 9 and 27.
You can see that this is a geometric sequence with common ratio 3 .
Sequences and series – Arithmetic Sequences
This type of sequence has been introduced briefly. Before looking into this in more detail there is some notation that we need to be aware of:
- First term: a_1=a
- Number of terms: n
- Last term: an=l
- Common difference: d
So if we take the arithmetic sequence 5, 7, 9, 11, 13, 15
a_1=5, l=17, d=2, n=6We can express the general term of an arithmetic sequence as follows:
a_k=a+(k-1) dAnd an expression for the last term can be written as:
l=a+(n-1) dExample
Find the 17th terms of the arithmetic sequence 12, 9, 6….
Solution
Here a=12, d= -3
Using a_k=a+(k-1) d we obtain:
a_{17}=12+(17-1)(-3)=-36Example
How many terms are in the sequence 11, 15, 19,…..,643?
Solution
Here a=11, d=4, l=643
Using l=a+(n-1) d we obtain:
\begin{gathered} 643=11+(n-1)(4) \\ \frac{643-11}{4}+1=n \\ n=159 \end{gathered}The sum of an arithmetic sequence
If you wish to find the sum of an arithmetic series then the following formula can be used:
S_n=\frac{n}{2}[2 a+(n-1) d]Example
Find the sum of the first 100 terms of the sequence 1, 1.25, 1.5, 1.75….
Solution
Here a=1, d=0.25, n=100
Using S_n=\frac{n}{2}[2 a+(n-1) d] we obtain the following:
\begin{gathered} S_{100}=\frac{100}{2}[2(1)+(100-1)(0.25)] \\ S_n=1337.5 \end{gathered}Example
The 5th term of an arithmetic sequence is 24 and the 9th term is 4. What is the first term and the common difference?
Solution
In this question we need to refer to the expression for the general term of an arithmetic sequence i.e. a_k=a+(k-1) d
k-5: a_5-a+4 d \text { i.e. } 24-a+4 dk=9: a_9=a+8 d \text { i.e. } 4=a+8 dWhat we have now is a set of simultaneous equations which can be solved.
\begin{aligned} & -20=4 d \text { so } d=-5 \\ & \therefore a=4-8 d=4-8(-5)=44 \end{aligned}The topics of sequences and series also includes a subtopic known as geometric series and we will cover this topic in another article.
Sequences and series is quite a nice A Level Maths topic and one that you should be able to get near if not full marks in the questions.