What a sequence is
A sequence is an ordered list of numbers. Each number in the list is called a term. Order matters: the first term, the second term and so on each have their own position.
- Sequence
- An ordered list of numbers built from a rule. Each number is a term, and the position of a term is its term number, \(n\).
For example, in the sequence \(4,\ 7,\ 10,\ 13,\ \ldots\) the first term is \(4\), the second term is \(7\), and the little dots mean the pattern keeps going. Our job is usually to find the rule and use it.
Term-to-term rules
The simplest way to describe a sequence is a term-to-term rule: what you do to one term to get the next one. Look at the gap or the multiplier between neighbours.
In \(4,\ 7,\ 10,\ 13,\ \ldots\) every term is \(3\) more than the one before, so the rule is "add \(3\)". In \(2,\ 6,\ 18,\ 54,\ \ldots\) every term is triple the one before, so the rule is "multiply by \(3\)". A rule can also subtract or divide.
Test the rule on more than one gap
Check the same rule works between every pair you can see, not just the first two. In \(4,\ 7,\ 10,\ 13\) the gaps are \(+3,\ +3,\ +3\): all equal, so "add \(3\)" is safe. If the gaps disagree, look for a multiply rule or a changing pattern instead.
Continuing a pattern
Once you know the term-to-term rule, continuing the sequence is just applying it again. Keep the rule steady and work along the list.
Write down the next two terms of \(4,\ 7,\ 10,\ 13,\ \ldots\)
Describing a sequence
To describe a simple sequence clearly you say two things: the first term and the term-to-term rule. Together they pin the sequence down completely.
So \(4,\ 7,\ 10,\ 13,\ \ldots\) is described as "start at \(4\) and add \(3\) each time". A sequence that goes up by a fixed amount is called arithmetic (or linear); one that multiplies by a fixed amount is called geometric. Naming the pattern and justifying it in words is the kind of clear communication credited under Criterion C, while spotting and using the rule is the reasoning of Criterion D.
Check yourself
1. Give the term-to-term rule and the next term of \(5,\ 9,\ 13,\ 17,\ \ldots\) +
The gaps are \(9-5=4\), \(13-9=4\), \(17-13=4\), so the rule is "add \(4\)". The next term is \(17 + 4 = \mathbf{21}\).
2. Describe the rule of \(2,\ 6,\ 18,\ 54,\ \ldots\) and write the next term +
Each term is three times the one before (\(6 = 2\times3\), \(18 = 6\times3\), \(54 = 18\times3\)), so the rule is "multiply by \(3\)". The next term is \(54 \times 3 = \mathbf{162}\).
3. Find the rule and the next term of \(20,\ 17,\ 14,\ 11,\ \ldots\) +
Each term is \(3\) less than the one before, so the rule is "subtract \(3\)". The next term is \(11 - 3 = \mathbf{8}\).
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