Linear Sequences and the nth Term
A linear (arithmetic) sequence is a list of numbers with a constant difference between consecutive terms. The nth term rule is an algebraic formula that gives any term directly from its position, n, without listing every term before it: for a linear sequence, the nth term takes the form an + b, where a is the common difference and b is found by comparing the sequence to the times table of a.
Before you start
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Method
- Find the common (constant) difference between consecutive terms by subtracting each term from the next.
- This common difference becomes the coefficient of n in the nth term rule, an + b.
- Find b by comparing the sequence to the times table of the common difference: work out what must be added to or subtracted from a x n to reach the first term when n = 1.
- Write the nth term rule in the form an + b.
- To find a specific term, substitute its position number for n into the rule.
- To check whether a given number is in the sequence, set the nth term rule equal to that number and solve for n; if n is a positive whole number, the value is in the sequence.
Worked example
Find the nth term rule for the sequence 5, 9, 13, 17, 21, ... and use it to find the 20th term.
- Find the common difference: 9 - 5 = 4, and 13 - 9 = 4, so the common difference is 4.
- The nth term rule starts with 4n, the 4 times table: 4, 8, 12, 16, 20, ...
- Compare this to the actual sequence, 5, 9, 13, 17, 21, ...: each term is 1 more than the 4 times table.
- The nth term rule is 4n + 1.
- Find the 20th term by substituting n = 20: 4 x 20 + 1 = 80 + 1 = 81.
Practice questions
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Q1Find the common difference of the sequence 3, 7, 11, 15, ...Show answer
Answer: 4
Q2Find the nth term rule for the sequence 2, 5, 8, 11, ...Show answer
Answer: 3n - 1
Q3Find the nth term rule for the sequence 6, 11, 16, 21, ...Show answer
Answer: 5n + 1
Q4Find the nth term rule for the sequence 10, 8, 6, 4, ...Show answer
Answer: 12 - 2n
Q5Using the nth term rule 4n - 3, find the 10th term.Show answer
Answer: 37
Q6Using the nth term rule 6n + 2, find the first three terms of the sequence.Show answer
Answer: 8, 14, 20
Q7Is 60 a term in the sequence with nth term rule 3n + 2? Explain your answer.Show answer
Answer: No; solving 3n + 2 = 60 gives 3n = 58, so n = 19.33, which is not a whole number, so 60 is not a term.
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
A sequence begins 4, 10, 16, 22, 28, ... (a) Find the nth term rule for this sequence. (b) Use your rule to find the 15th term.
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A sequence has nth term rule 5n + 3. Determine whether 98 is a term in this sequence, showing your working.
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Free printable worksheet
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This topic is chapter 30 of KS3 Maths Workbook 1, the whole course as one free printable PDF.
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