Algebra · GCSE Maths
Nth term of linear sequences
GCSE Maths linear sequences: find the common difference, write the nth term as an + b, and use it to generate terms or to decide whether a number is in the sequence.
A linear sequence goes up (or down) by the same amount each time. That common difference is a in an + b. Compare the first term with a × 1 to find b.
The important bits
What you need to know
- 1
A linear (arithmetic) sequence has a constant common difference d. 4, 7, 10, 13, … has d = 3. 20, 17, 14, … has d = −3.
- 2
The nth term is a + (n − 1)d, which rearranges to dn + (a − d). GCSE usually wants the second form: 3n + 1, 5n − 2, −2n + 9.
- 3
Method: find d. Write d n. Then find what you must add or subtract so that n = 1 recovers the first term. For 4, 7, 10, d = 3, 3n at n = 1 is 3, first term is 4, so add 1: 3n + 1.
- 4
To generate terms, substitute n = 1, 2, 3, … . n starts at 1 unless the question says otherwise. The 10th term of 3n + 1 is 31.
- 5
To test whether 100 is in the sequence 3n + 1, solve 3n + 1 = 100, 3n = 99, n = 33. If n is a positive integer, 100 is in the sequence.
- 6
If n is not an integer, the number is not in the sequence. 3n + 1 = 20 gives n = 19/3, so 20 is not a term.
- 7
The position-to-term rule is the nth term. A term-to-term rule such as “add 3” is not enough for the 100th term without a lot of adding.
- 8
Quadratic sequences (Higher) have a second difference that is constant. This page is the linear case: first differences constant, nth term degree 1.
Quotations worth analysing
Short evidence. Real method.
“nth term = an + b”
a is the common difference. b is chosen so that n = 1 gives the first term. Writing only “add 3” is a term-to-term rule and will not score as the nth term.
“Find the 50th term”
Substitute n = 50 into your rule; do not extend the list. The whole point of an + b is to jump to a distant term.
“Is 200 a term in the sequence?”
Set an + b = 200 and solve for n. Then state “n = … which is an integer, so yes” or “n is not an integer, so no”. The sentence is the mark.
Go deeper
Build an + b from the first term
Sequence: 5, 9, 13, 17. Differences: +4, +4, +4, so a = 4 and the sequence is “4n plus something”. When n = 1, 4n = 4, but the first term is 5, so we add 1: nth term = 4n + 1. Check n = 2: 8 + 1 = 9. n = 3: 12 + 1 = 13. A decreasing sequence: 11, 8, 5, 2. d = −3, so −3n plus something. At n = 1, −3n = −3, first term 11, so add 14: −3n + 14. Check: n = 2 gives −6 + 14 = 8. The check on two terms is not optional; it catches a b that was found from n = 0 by accident. Remember that n starts at 1 in the exam unless the question says otherwise.
Go deeper
Is this number in the sequence?
For 4n + 1, is 41 a term? 4n + 1 = 41, 4n = 40, n = 10, yes, the 10th term. Is 40 a term? 4n + 1 = 40, 4n = 39, n = 39/4, not an integer, no. Write that last sentence. A “show that 100 is not a term of 3n − 1” is the same algebra: 3n − 1 = 100, 3n = 101, n = 101/3, not an integer. Do not list twenty terms hoping 100 appears. If the sequence is of odd numbers, 2n − 1, every odd number works and n = (odd + 1)/2 is an integer; that is a useful check of the method. If a question gives a pattern of diagrams (matches, squares), count the extra matches each time — that extra is d — then find the nth term of the sequence of totals, not of the extras.
Go deeper
Term-to-term is not enough in the exam
“Add 7 each time” describes how to walk along the sequence but not how to land on the 80th term. The nth term is the position-to-term rule. Papers sometimes give the first term and the term-to-term rule and ask for the nth term: first term 6, add 5, so 5n + 1, because 5×1 + 1 = 6. Sometimes they ask for the term-to-term rule from the nth term: 5n + 1 increases by 5 each time. Know both directions. Linear versus quadratic: if the first differences change, but the second differences are constant, it is quadratic (an² + bn + c) and this page does not finish the job. If first differences are constant, stay linear. Mixing them, such as writing n² for a +3 sequence, is a topic-confusion error rather than an arithmetic one.
See the idea in action
The sequence is 4, 7, 10, 13, … Find the nth term, the 20th term, and whether 61 is in the sequence. Step 1: Common difference d = 7 − 4 = 3 (check 10 − 7 = 3). Step 2: Trial 3n: 3, 6, 9, 12, … which is 1 less than the sequence, so nth term = 3n + 1. Step 3: Check n = 1: 4. n = 4: 13. Both match. Step 4: 20th term = 3 × 20 + 1 = 61. Step 5: 3n + 1 = 61 gives 3n = 60, n = 20, which is a positive integer, so 61 is the 20th term. Contrast: 3n + 1 = 62 gives n = 61/3, not an integer, so 62 is not in the sequence.
Exam technique
Turn knowledge into marks
Write the differences under the terms, then 3n, then the adjustment to the first term. Substitute n = 1 and n = 2 before you use the rule for n = 50.
Common mistakes
Do not give these marks away
- 01
Offering “add 3” as the nth term, which is a term-to-term rule and does not give the 50th term.
- 02
Using n = 0 as the first term, so an + b is out by one position.
- 03
Deciding a number is in the sequence by listing a few terms, then guessing, instead of solving an + b = that number.
The nth term of 4, 7, 10, 13, … is
A3n + 1
B3n
C4n + 3
Dn + 3
Show the answer
3n + 1. Common difference 3, and 3n + 1 gives 4, 7, 10, 13. 3n gives 3, 6, 9, 12. 4n + 3 gives 7, 11, 15. n + 3 is 4, 5, 6, 7, which only matches the first term.
Quick questions
If this is the bit you searched
How do you find the nth term of a linear sequence?
Find the common difference a. Write an. Then choose b so that n = 1 gives the first term. Check on the second term.
How do I find the 100th term?
Substitute n = 100 into the nth term. Do not write out 100 terms. If you only have a term-to-term rule, find the nth term first.
How do I show a number is not in the sequence?
Set the nth term equal to the number, solve for n, and show that n is not a positive integer. That sentence is the conclusion.
What is the difference between a term-to-term rule and the nth term?
Term-to-term tells you how to get the next term from the one before (add 3). The nth term tells you the term in position n directly (3n + 1).