The big idea: To add up the first n terms of an arithmetic sequence, use the sum formula.
There are two forms — pick the one that matches what you know.
IB-style question — a simple sum
Find the sum of the first 10 terms of 4 + 7 + 10 + … .
Step by step
- Write the sum formula.
- Spot u₁ = 4, d = 3, n = 10 and substitute.
- Work it out.
Final answer
S₁₀ = 175.
Which form, when?: Know u₁ and d → use Sₙ = (n/2)(2u₁ + (n − 1)d).
Know u₁ and the last term uₙ → use Sₙ = (n/2)(u₁ + uₙ).
Reading the formula
- Sₙ means the sum of the first n terms.
- The (n/2) is half the number of terms.
- The bracket adds the first and last term (bottom form).
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Pick the matching form: Both forms find Sₙ — the sum of the first n terms.
Given u₁, d and n, use the first form. Given u₁, the last term and n, the second form is faster.
IB-style question — know u₁ and d
An arithmetic sequence has u₁ = 6 and d = 4.
Find the sum of the first 10 terms.
Step by step
- Choose the form for u₁ and d.
- Substitute u₁ = 6, d = 4, n = 10.
- Simplify inside, then multiply.
Final answer
S₁₀ = 240.
IB-style question — know the first and last term
The first term of an arithmetic sequence is 6 and the 20th term is 82.
Find the sum of the first 20 terms.
Step by step
- Choose the form for the first and last term.
- Substitute u₁ = 6, u₂₀ = 82, n = 20.
- Simplify.
Final answer
S₂₀ = 880.
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When Sₙ is given as a formula: Sometimes you are told Sₙ is a quadratic in n, like Sₙ = 2n² + 3n.
Two facts unlock everything: u₁ = S₁ and uₙ = Sₙ − Sₙ₋₁.
IB-style question — terms from a given sum
The sum of the first n terms of an arithmetic sequence is Sₙ = 2n² + 3n.
Find u₁ and a general expression for uₙ.
Step by step
- The first term is just S₁.
- For any term use uₙ = Sₙ − Sₙ₋₁. First write Sₙ₋₁ by putting (n − 1) in place of every n.
- Expand each bracket, then tidy up.
- Subtract: Sₙ − Sₙ₋₁. The 2n² terms cancel, leaving a linear term.
Final answer
u₁ = 5 and uₙ = 4n + 1 (check: u₁ = 5 ✓).
IB-style question — combine a term and a sum
In an arithmetic sequence u₅ = 20 and S₅ = 70.
Find u₁ and the common difference.
Step by step
- Write the first-and-last form (you know S₅ and u₅).
- Substitute n = 5, u₅ = 20, S₅ = 70.
- A fraction in the way? Multiply both sides by the bottom (2) to clear it.
- Expand the bracket.
- Subtract 100, then divide by 5.
- You're also told a term, u₅ = 20 — so use the term formula uₙ = u₁ + (n − 1)d with n = 5 (n − 1 = 4).
- Put in u₁ = 8 and u₅ = 20, then solve.
Final answer
u₁ = 8 and d = 3.
Common mistakes
- Mixing up Sₙ (a sum) with uₙ (a single term).
- Multiplying by n instead of n/2.
- When Sₙ is quadratic, forgetting uₙ = Sₙ − Sₙ₋₁.
Do this instead
- Sₙ is a running total; uₙ is one term.
- The formula has n/2 — half the number of terms.
- u₁ = S₁ and uₙ = Sₙ − Sₙ₋₁ recover every term.
Which paper?: This is the most common Unit-1 Paper-1 question — usually 5–7 marks.
The maximum-sum version appears on Paper 2 (see Applications).