Arithmetic sequences and series
An arithmetic sequence goes up or down by the same amount each time. Two formulas, one for any term and one for a sum, cover almost every question, including savings plans and stacked objects.
Part 1 of 3: Learn it
In short
- uₙ = a + (n − 1)d, where a is the first term and d the common difference.
- Sₙ = n/2 × (2a + (n − 1)d), or n/2 × (a + l) if you know the last term l.
- Σ notation adds up terms: Σ from r = 1 to n of uᵣ.
Where this is in your specification
Spec points: DfE D6 and D8 (AQA 7357 D6 and D8, Edexcel 9MA0 Pure topic 4, OCR A H240 1.04)
| Board | Topic: Sequences, series and the binomial expansion |
|---|---|
| DfE content | D1-D8 |
| AQA 7357 | D1-D8 |
| Edexcel 9MA0 | Pure topic 4 |
| OCR H240 | 1.04 |
| Higher C847 76 | Algebraic and trigonometric skills: recurrence relations |
The nth term
The (n − 1) is there because the first term has had no differences added yet. If two terms are given, write an equation for each and solve them simultaneously to find a and d.
The sum of n terms
The second form comes from pairing the first and last terms, the second and second-last, and so on: each pair has the same total. That is how 1 + 2 + ... + 100 = 50 × 101 = 5050.
Sigma notation
Σ (r = 1 to 10) of (2r + 1) means put r = 1, 2, ..., 10 into 2r + 1 and add: 3 + 5 + ... + 21. It is arithmetic with a = 3, d = 2, n = 10, so the sum is 5 × (3 + 21) = 120.
What is the common difference of 7, 4, 1, −2, ...?
Show the answer
−3.
Part 2 of 3: See it worked
Worked examples
Example 1
For 5, 9, 13, 17, ..., find the 20th term and the sum of the first 20 terms.
- a = 5, d = 4: u₂₀ = 5 + 19 × 4 = 81
- S₂₀ = ½ × 20 × (5 + 81) = 10 × 86
Answer: u₂₀ = 81 and S₂₀ = 860.
Example 2
Sam saves £50 in the first month and £10 more each month than the month before. How much has Sam saved in total after 24 months?
- a = 50, d = 10, n = 24
- S₂₄ = ½ × 24 × (2 × 50 + 23 × 10)
- = 12 × 330
Answer: £3960.
Common mistakes
- Using n instead of n − 1 in the nth term formula.
- Giving a term when the question asks for a sum, or the other way round.
- Forgetting that d is negative for a decreasing sequence.
- Counting the number of terms wrongly in sigma notation, especially when r does not start at 1.
Find the sum of the whole numbers from 1 to 100.
Show the answer
½ × 100 × (1 + 100) = 5050.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
What is the common difference of 7, 4, 1, −2, ...?
−3.
Find the sum of the whole numbers from 1 to 100.
½ × 100 × (1 + 100) = 5050.
An arithmetic sequence has a = 2 and d = 3. Find u₁₅.
2 + 14 × 3 = 44.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
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