The sum of an arithmetic series is the total of all terms in a sequence where consecutive terms differ by a fixed amount. The shortcut was discovered by Gauss: pair the first and last terms, multiply their sum by half the number of terms. Two equivalent formulae follow, and both are worth knowing for GCSE Higher questions.
What is an arithmetic sequence and an arithmetic series?
An arithmetic sequence is an ordered list of numbers with a constant difference between successive terms, called the common difference d.
Examples:
- 3, 7, 11, 15, 19, … (d = 4)
- 20, 17, 14, 11, 8, … (d = −3)
An arithmetic series is the sum of the terms in such a sequence. Adding the first four terms of the first example gives 3 + 7 + 11 + 15 = 36.
For short sequences, adding directly is fine. For longer ones — "find the sum of the first 100 terms" — a formula is essential.
How did Gauss derive the sum formula?
The story goes that Carl Friedrich Gauss, aged about nine, was asked to add 1 + 2 + 3 + … + 100. He noticed that pairing the first and last terms gives 1 + 100 = 101; the second and second-to-last gives 2 + 99 = 101; and so on, producing 50 pairs each summing to 101.
Sum = 50 × 101 = 5050
In general, an arithmetic sequence with first term a, last term l, and n terms can be paired into n/2 pairs each with sum (a + l):
Sₙ = (n/2)(a + l)
Since the last term l = a + (n − 1)d, substituting gives the second form:
Sₙ = (n/2)(2a + (n − 1)d)
Both formulae give the same result; use whichever suits the information given.
How do you use the formula when you know the first and last terms?
Use Sₙ = (n/2)(a + l) when you know the first term a, the last term l, and the number of terms n.
Worked example: Find the sum of the arithmetic series 5 + 9 + 13 + … + 49.
- Identify a = 5, l = 49, d = 4.
- Find n: the nth term formula gives a + (n − 1)d = l, so 5 + (n − 1) × 4 = 49. (n − 1) × 4 = 44 → n − 1 = 11 → n = 12.
- Apply the formula: S₁₂ = (12/2)(5 + 49) = 6 × 54 = 324.
How do you use the formula when you know the first term and common difference?
Use Sₙ = (n/2)(2a + (n − 1)d) when you know a, d and n but not the last term.
Worked example: Find the sum of the first 20 terms of the sequence 3, 7, 11, 15, …
- a = 3, d = 4, n = 20.
- Sₙ = (20/2)(2 × 3 + (20 − 1) × 4) = 10 × (6 + 76) = 10 × 82 = 820.
Check using the last term: 20th term = 3 + 19 × 4 = 79. S₂₀ = (20/2)(3 + 79) = 10 × 82 = 820 ✓
What does each formula symbol mean?
| Symbol | Meaning | Example |
|---|---|---|
| n | Number of terms being added | 20 |
| a | First term of the sequence | 3 |
| d | Common difference | 4 |
| l | Last term in the sum | 79 |
| Sₙ | Sum of the first n terms | 820 |
How do you find n or d from a given sum?
If the sum Sₙ is given, substitute it and the known values into the formula, then solve algebraically.
Worked example: The sum of the first n terms of an arithmetic sequence is 120. The first term is 4 and the common difference is 2. Find n.
Sₙ = (n/2)(2a + (n − 1)d)
120 = (n/2)(8 + 2(n − 1))
120 = (n/2)(2n + 6)
240 = n(2n + 6) = 2n² + 6n
2n² + 6n − 240 = 0
n² + 3n − 120 = 0
Using the quadratic formula: n = (−3 ± √(9 + 480)) / 2 = (−3 ± √489) / 2 ≈ (−3 ± 22.1) / 2.
Taking the positive root: n ≈ 19.1/2 ≈ 9.55 — not a whole number, so check the problem values. (If the question is set correctly, n will be a positive integer; if it is not, recheck the substitution.)
Frequently asked questions
What is the difference between a sequence and a series?
A sequence is an ordered list: 2, 5, 8, 11, … A series is the sum of the terms: 2 + 5 + 8 + 11 + … Sequences list; series add. In everyday language the words are often used interchangeably, but in GCSE maths they have distinct meanings.
Can the formula be used if the common difference is negative?
Yes — a decreasing sequence such as 30, 27, 24, … has d = −3. Substitute the negative value for d in either formula. The sum will be smaller than n × a, and for a long enough sequence may even become negative if the terms go below zero.
How do I know which of the two formulae to use?
If the question gives you the last term l, use Sₙ = (n/2)(a + l) — it is quicker. If you only know a, d and n, use Sₙ = (n/2)(2a + (n − 1)d). If you have all five quantities (a, d, n, l, Sₙ), either formula works and they will give the same answer.
Does this formula appear on the GCSE formula sheet?
No — the sum formula for arithmetic series is not provided on AQA or Edexcel GCSE formula sheets. It must be memorised. However, many Higher questions ask you to derive or use it, so understanding the Gauss pairing argument (rather than just memorising the formula) helps you reconstruct it if needed.
For GCSE Higher algebra coaching including sequences and series, visit aitutors.me.