A sequence is a set of numbers that all follow a rule. You need to know about two different types of sequences arithmetic and geometric sequences.
Arithmetic sequences
An arithmetic sequence is a sequence that has a common difference, and this means that the sequence will either increase or decrease by a constant addition or subtraction. They look like this:
3, 7, 11, 15, 19 ... This sequence has a common difference of 4
78, 72, 66, 60, 54 ... This sequence has a common difference of 6
5, 12, 19, 26, 33 ... This sequence has a common difference of 7
You may need to find a specific term (nth term) within the sequence and to do this you can use this formula;
\[u_n = a + (n-1)d\]
un is the nth term a is the first term d is the common difference
Find the 50th term of the following sequence 4, 7, 10, 13, 16, 19 ...
First, you need to identify your variables and substitute them into the formula;
n - 50
a - 4
d - 3
\[u_{50} = 4 + (50-1)3\]
Now you need to solve the equation.
\[u_{50} = 4 + (50-1)3\]
\[u_{50} = 151\]
Geometric sequences
A geometric sequence is a sequence that has a common ratio, the sequence will either increase or decrease by a constant multiplication or division. Here are some examples:
- 3, 9, 27, 81, 243 ... This sequence has a common ratio of 3
- 9, 18, 36, 72, 144 ... This sequence has a common ratio of 2
- 4, 6, 9, 13.5, 20.25 ... This sequence has a common ratio of 1.5
You may also be asked to find a specific term from this sequence, below is the formula that you would need;
\[u_n = ar^{n-1}\]
un is the nth term a is the first term r is the common ratio
Find the 15th term of this sequence 1, 2, 4, 8, 16 ...
First you need to identify your variables and substitute them into the formula;
n - 15
a - 1
r - 2
\[u_{15} = (1)2^{15-1}\]
Now you solve your equation.
\[u_{15} = (1)2^{15-1}\]
\[u_{15} = 16384\]
Recurrence relations
You are able to find each term of the sequence if you know the rule that it is following and the first term using a recurrence relation. You can use each previous term to help you find the next one, and the formula for this is;
\[u_{n+1} = f(u_n)\]
You may be given this function and asked to find the first number of terms. Let's have a look at how you would approach this type of question;
Find the next five terms of the sequence \(u_{n+1} = u_n + 3, u_1 = 7\)
To do this, you need to substitute the nth term into the formula;
term 1 - \(u_2 = u_1 +3\) \(u_2 = 7 + 3\) \(u_2 = 10\)
term 2 - \(u_3 = u_2 +3\) \(u_3 = 10 + 3\) \(u_3 = 13\)
term 3 - \(u_4 = u_3 +3\) \(u_4 = 13 + 3\) \(u_4 = 16\)
term 4 - \(u_5 = u_4 +3\) \(u_5 = 16 + 3\) \(u_5 = 19\)
term 5 - \(u_6 = u_5 +3\) \(u_6 = 19 + 3\) \(u_6 = 122\)
Increasing and decreasing sequences
Sequences can be described as increasing if each term is higher than the previous one, this can be shown as, \(u_{n+1} > u_n\). They can be described as decreasing if each term is less than the previous one, this can be shown as \(u_{n+1} < u_n\). A sequence can also be described as periodic if the terms within the sequence repeat, or create a cycle, this can be shown as \(u_{n+k} = u_n\).
An example of an increasing sequence 7, 15, 23, 31, 39, 47
An example of a decreasing sequence 15, 10, 5, 0, -5, -10
An example of a periodic sequence 8, 9, 10, 8, 9, 10, 8, 9, 10
How to model real-life scenarios with sequences
Sequences can be used to model many real-life scenarios, such as savings and salaries. If the model increases by the same amount, it will create an arithmetic sequence; if it increases by the same percentage, it will create a geometric sequence.
A woman has £2000 in her savings account, and each month, she adds £200. How much money would she have in her savings account after one year?
Let's break down the question. First, we need to identify the type of sequence that it is. As the constant increases by the same amount each month, it is an arithmetic sequence. Next, we need to find the correct formula to use to help us find how much money is in the account after 1 year, meaning you need to find the 12th term;
\[u_n = a + (n-1)d\]
Next you need to substitute in the information that you know
a - 2000
n - 12
d - 200
\[u_{12} = 2000 + (12-1)200\]
Now solve the equation that you have created.
\[u_{12} = 2000 + (12-1)200\]
\[u_{12} = 4200\]
You now know that the woman will have £4200 in her savings account after 12 months.
Sequences - key takeaways
A sequence is a set of numbers that follow a specific rule and order.
There are two types of sequences, arithmetic and geometric.
An arithmetic sequence increases and decreases by addition and subtraction.
A geometric sequence increases and decreases by multiplication and division.
You can use a formula to find a specific term within the sequence.
Sequences can be used to model real life scenarios.
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