A linear (arithmetic) sequence goes up or down by the same amount each time, called the common difference. Its nth term is the common difference × n plus a number that makes the first term correct: for 5, 9, 13, 17, …, the difference is 4 and the nth term is 4n + 1. Substitute a value of n to find any term, and solve an equation to test whether a number is in the sequence.
Linear sequences
- Common difference: the amount added each time
- nth term = (common difference) × n + (first term − common difference)
- Find a term: substitute n
- Is k in the sequence? Solve nth term = k; it is a term only if n is a positive whole number
Term-to-term rules
A term-to-term rule tells you how to get from one term to the next, such as “add 5” or “subtract 6”. For 6, 11, 16, 21, … the rule is add 5, so the next two terms are 26 and 31. Term-to-term rules are easy to continue but slow for finding, say, the 100th term.
Finding the nth term
The nth term is a position-to-term rule. For a linear sequence, the common difference is the coefficient of n. Then work out what to add or subtract so the first term is correct. For 5, 9, 13, 17, …, the difference is 4, giving 4n; when n = 1, 4n = 4, but the first term is 5, so add 1: the nth term is 4n + 1.
Decreasing sequences
If the sequence goes down, the common difference is negative. For 40, 34, 28, 22, …, the difference is −6. When n = 1, −6n = −6, and the first term is 40, so add 46: the nth term is −6n + 46, often written 46 − 6n.
Using the nth term
Substitute a position to find a term: with nth term 4n + 1, the 50th term is 4 × 50 + 1 = 201. To test whether a number is in the sequence, set the nth term equal to it and solve. 4n + 1 = 75 gives n = 18.5, which is not a whole number, so 75 is not in the sequence.
Other types of sequence
Not every sequence is linear. A geometric sequence multiplies by the same number each time, such as 3, 6, 12, 24, … In a Fibonacci-type sequence each term is the sum of the two before. Square numbers, 1, 4, 9, 16, …, and triangular numbers, 1, 3, 6, 10, …, also appear.
Worked examples
Example 1. Write down the next two terms of 6, 11, 16, 21, …
- The sequence goes up by 5 each time.
- 21 + 5 = 26, and 26 + 5 = 31.
Answer: 26, 31.
Example 2. Find the nth term of 5, 9, 13, 17, …
- The common difference is 4, so start with 4n.
- When n = 1, 4n = 4; the first term is 5, so add 1.
Answer: 4n + 1.
Example 3. Work out the 50th term of the sequence with nth term 4n + 1.
- Substitute n = 50: 4 × 50 + 1.
Answer: 201.
Example 4. Is 75 a term in the sequence with nth term 4n + 1? Show your working.
- Solve 4n + 1 = 75: 4n = 74, so n = 18.5.
- 18.5 is not a whole number.
Answer: No, 75 is not a term.
Example 5. Find the nth term of 40, 34, 28, 22, …
- The common difference is −6, so start with −6n.
- When n = 1, −6n = −6; the first term is 40, so add 46.
Answer: 46 − 6n.
Common mistakes
- Using the first term as the coefficient of n. The coefficient of n is the common difference.
- Forgetting the adjustment. Check your nth term gives the correct first term.
- Accepting non-integer positions. A number is only in the sequence if n is a positive whole number.
- Getting the sign wrong for decreasing sequences. A decreasing sequence has a negative coefficient of n.
- Assuming every sequence is linear. Check that the difference really is constant.
How sequences are examined
Linear sequences are taught at Foundation tier for both Edexcel specifications, 1MA1 and 4MA1, and Higher tier goes on to quadratic sequences. Typical questions ask you to:
- continue a sequence and describe its term-to-term rule
- find the nth term of a linear sequence
- work out a particular term from the nth term
- decide whether a number is in a sequence, with working
Use the sequences worksheet, which has answers, and the GCSE algebra revision engine for more.
At A-Level, linear sequences become arithmetic sequences. Finding the nth term uses substitution.
Frequently asked questions
What is the method for the nth term of a linear sequence?
Use the common difference as the coefficient of n, then add or subtract a number so the first term is correct.
How do you know if a number is in a sequence?
Set the nth term equal to the number and solve. If n is a positive whole number, it is in the sequence.
What is a term-to-term rule?
A rule that tells you how to get from one term to the next, such as “add 3”.