Have you ever wondered how you may find out what your salary may be in 5 years if it is increasing by a constant amount each year? Jake earns an annual salary of £27,000, let's call this 'x' and has an annual increment of let's call this 'y'. To find out Jake's annual salary for the coming 3 years you can use the expression x+y, x+2y, x+3y. This is an example of a sequence. It can be helpful to understand how to generate terms of a sequence for many real-life reasons.
General term of a sequence definition
A sequence is a set of numbers that all follow a certain pattern or rule.
There are different types of sequences, such as arithmetic and geometric sequences.
An arithmetic sequence is a type of sequence that increases or decreases by a constant addition or subtraction. This is known as a common difference.
A geometric sequence is a type of sequence that increases or decreases by a constant multiplication or division. This is known as the common ratio.
Generate terms of a sequence calculations
It can be helpful to be able to generate terms within a sequence. This can be done using the term-to-term rule or the position-to-term rule.
Term-to-term rule
The term-to-term rule is a way in which the terms of a sequence are increasing or decreasing. To use this rule you must find the common difference or the common ratio depending on the type of sequence you have. Once you understand how to find the common difference/ratio, you can use the term-to-term rule to generate terms of a sequence.
Arithmetic sequences
To use the term-to-term rule within an arithmetic sequence you need to first calculate the common difference between one term and the next, this is usually done by subtracting one term from the previous term.
Find the common difference for 3, 7, 11, 15, 19...
Solution:
To start with you can subtract the second term from the first term, then check that the difference is constant by checking the difference between each of the terms.
Here you can see that each of the terms has a difference of 4, therefore the common difference is 4.
Geometric sequences
To find the term-to-term rule of a geometric sequence you need to divide one term by the previous term.
Find the common ratio for 5, 10, 20, 40...
Solution:
To find the common ratio you can divide one term by the previous term, then you can check that the difference between each term is constant;
Here you can see that the common ratio is 2.
Using the term-to-term rule to generate terms of a sequence
To use the term-to-term rule to find the next terms of a sequence, you can simply calculate the common difference/ratio and then use that number to find the following terms.
Use the term-to-term rule to find the next 3 terms of the following sequence; 6, 11, 16, 21...
Solution:
Find the common difference by calculating the difference between each term to ensure it is constant.
You can see here that the common difference is 5, since you know this you can now use this information to find the next terms of the sequence. This is done by adding 5 to the last term to find the next term.
Therefore, for the sequence; 6, 11, 16, 21... the common difference is 5 and the next three terms are 26, 31 and 36.
Position to term rule
The position to term rule refers to the position each term is in, as each term has its own position. This rule can be used to create a formula for a sequence, which will allow you to generate terms of a sequence. Within the formula, it can be written in terms of n. The nth term represents a particular term within a sequence.
Calculating the position to term rule
Since each term within a sequence has a position you are able to calculate this and use this to create a formula for the sequence.
Let's look at the sequence; 10, 11, 12, 13, 14...
Solution:
To visually show how to identify the position of each of these terms we will put them into a table.
| Position | Term |
| 1 | 10 |
| 2 | 11 |
| 3 | 12 |
| 4 | 13 |
| 5 | 14 |
In order to create the formula, you need to consider how the terms are getting from the position to the term, for example, to get from position 1 to the term 10 you add 9. Again, to get from position 2 to term 11, you add 9 and so on. This means that the formula will be written as
Using the position to term rule to generate terms of a sequence
In order to use the position to term rule to generate terms of a sequence, you can substitute the nth term into the given formula, this will give you the term for that position.
Find the first three terms of the sequence where
Solution:
Since you need to find the first three terms of the sequence, you simply substitute the positions, 1, 2 and 3 into the formula to find the value of the terms;
Therefore the first three terms of the sequence are 8, 10, 12.
General term of a sequence examples
Find the first three terms of the sequence where using the position to term rule.
Solution:
Since you need to find the first three terms of the sequence, you simply substitute the positions into the formula.
Therefore the first three terms of the sequence are 2, 6, 10.
Use the term-to-term rule to find the next 3 terms of the following sequence; 4, 12, 36, 108...
Solution:
Start by finding the constant between each other the terms:
You can see here that the common ratio is 3, now you know this you can use this information to find the next terms of the sequence. This is done by multiplying the last term by 3 to find the next term;
Therefore the next 3 terms of the sequence are 324, 972 and 2916.
Generating Terms of a Sequence - Key takeaways
- A sequence is a set of numbers that all follow a certain pattern or rule.
- To generate terms within a sequence you can use the term-to-term rule or the position-to-term rule.
- The term-to-term rule is a way of describing the way in which the terms in a sequence are increasing or decreasing. In order to generate the following term of a sequence, you can add, subtract, divide or multiply the last term by the term-to-term rule.
- The position to term rule refers to the position each term is in and can help generate a formula that will allow you to generate terms of a sequence.
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