Describing Sequences
Describing Sequences
A sequence is an ordered list of items, usually numbers. Each item which makes up a sequence is called a “term”.
Sequences can have interesting patterns. Here we examine some types of patterns and how they are formed.
Examples:
-
\(1; 4; 7; 10; 13; 16; 19; 22; 25; \ldots\)
There is difference of \(\text{3}\) between successive terms.
The pattern is continued by adding \(\text{3}\) to the previous term.
-
\(13; 8; 3; -2; -7; -12; -17; -22; \ldots\)
There is a difference of \(-\text{5}\) between successive terms.
The pattern is continued by adding \(-\text{5}\) to (i.e. subtracting \(\text{5}\) from) the previous term.
-
\(2; 4; 8; 16; 32; 64; 128; 256; \ldots\)
This sequence has a factor of \(\text{2}\) between successive terms.
The pattern is continued by multiplying the previous term by 2.
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\(3; -9; 27; -81; 243; -729; 2187; \ldots\)
This sequence has a factor of \(-\text{3}\) between successive terms.
The pattern is continued by multiplying the previous term by \(-\text{3}\).
-
\(9; 3; 1; \cfrac{1}{3}; \cfrac{1}{9}; \cfrac{1}{27}; \ldots\)
This sequence has a factor of \(\cfrac{1}{3}\) between successive terms.
The pattern is continued by multiplying the previous term by \(\cfrac{1}{3}\) which is equivalent to dividing the previous term by 3.
Example
Question
You and \(\text{3}\) friends decide to study for Maths and are sitting together at a square table. A few minutes later, \(\text{2}\) other friends arrive and would like to sit at your table. You move another table next to yours so that \(\text{6}\) people can sit at the table. Another \(\text{2}\) friends also want to join your group, so you take a third table and add it to the existing tables. Now \(\text{8}\) people can sit together.
Examine how the number of people sitting is related to the number of tables. Is there a pattern?
Figure 3.1: Two more people can be seated for each table added.
Make a table to see if a pattern forms
|
Number of tables, n |
Number of people seated |
|
\(\text{1}\) |
\(4=4\) |
|
\(\text{2}\) |
\(4+2=6\) |
|
\(\text{3}\) |
\(4+2+2=8\) |
|
\(\text{4}\) |
\(4+2+2+2=10\) |
|
\(\vdots\) |
\(\vdots\) |
|
n |
\(4+2+2+2+\cdots +2\) |
Describe the pattern
We can see that for \(\text{3}\) tables we can seat \(\text{8}\) people, for \(\text{4}\) tables we can seat \(\text{10}\) people and so on. We started out with \(\text{4}\) people and added two each time. So for each table added, the number of people increased by \(\text{2}\).
So the pattern formed is \(4; 6; 8; 10; \ldots\).
To describe terms in a number pattern we use the following notation:
The first term of a sequence is \({T}_{1}\).
The fourth term of a sequence is \({T}_{4}\).
The tenth term of a sequence is \({T}_{10}\).
The general term is often expressed as the \(n^{\text{th}}\) term and is written as \({T}_{n}\).
A sequence does not have to follow a pattern but, when it does, we can write down the general formula to calculate any term. For example, consider the following linear sequence:
\(1; 3; 5; 7; 9; \ldots\)The \(n^{\text{th}}\) term is given by the general formula:
\({T}_{n}=2n-1\)You can check this by substituting values into the formula:
\begin{align*} {T}_{1}& = 2(1)-1=1 \\ {T}_{2}& = 2(2)-1=3 \\ {T}_{3}& = 2(3)-1=5 \\ {T}_{4}& = 2(4)-1=7 \\ {T}_{5}& = 2(5)-1=9 \end{align*}If we find the relationship between the position of a term and its value, we can find a general formula which matches the pattern and find any term in the sequence.
This lesson is part of:
Sequences and Series