Sequences: Term-to-Term and Position-to-Term
Sequences and their term-to-term and position-to-term rules are a core 11+ maths topic. A sequence is a list of numbers or shapes following a pattern; a term-to-term rule describes how to get from one term to the next, such as 'add 3', while a position-to-term rule (the nth term rule) gives a formula to calculate any term directly from its position in the sequence, without working out every term before it. 11+ papers test finding both kinds of rule, continuing a sequence, and finding a specific term such as the 20th term.
Method
- To find a term-to-term rule, calculate the difference between consecutive terms; if the difference is constant, that difference is the rule.
- To continue a sequence, keep applying the term-to-term rule to the last known term.
- To find the position-to-term (nth term) rule for a linear sequence, use the constant difference as the coefficient of n, then adjust the constant by comparing to the actual first term.
- To find a specific term, such as the 20th term, using the nth term rule, substitute the position number in place of n and calculate.
- Watch for sequences that are not linear, such as those that double each time, or that follow a shape-based pattern like triangular numbers; these need their pattern identified separately.
- Check your rule works by testing it against at least two terms you already know from the sequence.
Worked example
A sequence begins 4, 9, 14, 19, 24. Find the nth term rule for this sequence, and use it to find the 15th term.
- Find the term-to-term difference: 9 - 4 = 5, 14 - 9 = 5, so the sequence increases by 5 each time.
- Since the difference is constant at 5, the nth term rule starts with 5n.
- Compare 5n to the actual first term: when n = 1, 5n = 5, but the first term is 4, which is 1 less than 5.
- So the nth term rule is 5n - 1.
- Find the 15th term by substituting n = 15: 5 x 15 - 1 = 75 - 1 = 74.
Practice questions
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Q1Find the next two terms of the sequence 6, 11, 16, 21, ...Show answer
Answer: 26, 31
Q2Find the term-to-term rule for the sequence 40, 34, 28, 22, ...Show answer
Answer: Subtract 6 each time
Q3Find the nth term rule for the sequence 3, 6, 9, 12, ...Show answer
Answer: 3n
Q4Find the nth term rule for the sequence 7, 10, 13, 16, ...Show answer
Answer: 3n + 4
Q5Using the nth term rule 4n - 3, find the 10th term of the sequence.Show answer
Answer: 37 (4 x 10 - 3)
Q6A sequence follows the rule 'double the previous term, then subtract 1', starting at 3. Find the first four terms.Show answer
Answer: 3, 5, 9, 17
Q7The nth term of a sequence is 2n + 5. Which term of the sequence is equal to 23?Show answer
Answer: The 9th term (2n + 5 = 23 gives n = 9)
Exam-style questions
Written in the style of a 11+ exam paper, with a full mark scheme.
A sequence begins 5, 12, 19, 26, 33. (a) Find the term-to-term rule. (b) Find the nth term rule. (c) Use your rule to find the 25th term.
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The nth term rule of a sequence is 6n + 1. (a) Find the first three terms of the sequence. (b) Explain, using your rule, whether 85 is a term in this sequence.
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Free printable worksheet
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This topic is chapter 18 of 11+ Maths Workbook 2, the whole course as one free printable PDF.
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