GCSE-readyArithmetic sequences and series
International GCSE Higher uses formulae for arithmetic sequences: the nth term and the sum of the first n terms.
The nth term
nth term = a + (n − 1)d
4, 7, 10, ...: a = 4, d = 3, so the 15th term is 4 + 14 × 3 = 46.
The sum of n terms
Sum of the first 20 terms of 4 + 7 + 10 + ...:
S20 = 10[8 + 19 × 3] = 10 × 65 = 650
Finding a, d or n
If the 3rd term is 11 and the 8th term is 26: a + 2d = 11 and a + 7d = 26.
Subtract: 5d = 15, so d = 3 and a = 5.
Find the sum 3 + 7 + 11 + ... + 79.
- a = 3, d = 4. 3 + (n − 1)4 = 79 gives n = 20.
- S = 202(2 × 3 + 19 × 4)
- = 820
Answer: 820
nth term a + (n − 1)d. Sum Sn = n/2[2a + (n − 1)d]. Two facts give two equations in a and d.
- nth term
- A rule that gives any term of a sequence from its position n, such as 3n + 2.
- Term
- A single number, letter or product of them in an expression, such as 5x or −3.
- Arithmetic sequence
- A sequence where you add the same number (the common difference) each time, such as 5, 8, 11, 14.
All GCSE and International GCSE Mathematics key terms and definitions →
2 more questions
Try each one, then tap to check.
Find the 15th term of 12, 17, 22, ...
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Answer: 82
12 + 14 × 5 = 82.
Find the sum of the first 25 terms of 11 + 20 + 29 + ...
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Answer: 2975
252(22 + 216) = 2975.
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- Pearson Edexcel International GCSE Mathematics (4MA1) specification, topic 3.1
- Pearson Edexcel International GCSE Mathematics A (4MA1) specification, Appendix 5 Higher formulae sheet
Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.