Key Idea: An arithmetic sequence changes by the same fixed amount every step. That amount is d, the common difference.
Three things IB tests on this topic:
⚡ Sequence vs Series — do not mix these up
Tip: Asked for the 10th term? → Use uₙ. Asked for the total or sum? → Use Sₙ. These are completely different things. One is a single value, one is a running total.
✏️ Worked examples
IB-style question — write down d
A concert hall sells tickets with prices that form an arithmetic sequence. The closer to the stage, the more expensive the seat. Row 1: €480 Row 2: €460 Row 3: €440 (a) Write down the value of the common difference, d.
Step by step:
Subtract any term from the one after it:
d = 460 − 480 = −20
Negative d means seats get cheaper the further from the stage.
d = −20
IB-style question — find uₙ
(b) Calculate the price of a ticket in row 16.
Step by step:
Use uₙ = u₁ + (n − 1)d with u₁ = 480, d = −20, n = 16:
u₁₆ = 480 + (16 − 1) × (−20)
u₁₆ = 480 − 300 = €180
€180
IB-style question — find the total (with multiplier)
(c) Find the total cost of buying 2 tickets in each of the first 16 rows.
Step by step:
Find the sum of one ticket per row using Sₙ = n/2 (u₁ + uₙ):
S₁₆ = 16/2 × (480 + 180) = 8 × 660 = €5 280
2 tickets per row → multiply by 2:
Total = 5 280 × 2 = €10 560
Watch out: the question says 2 tickets per row — easy to miss the × 2.
€10 560
∑ Sigma notation — read each part separately
Read each part of the sigma separately:
BOTTOM
The bottom tells you where to start.
→ Start at n = 1
TOP
The top tells you where to stop.
→ Stop at n = 4
EXPRESSION
The expression on the right tells you what to add each time.
→ Add 2n + 1 for each n
How to evaluate sigma notation
Evaluate this sigma expression:
Step by step:
Plug in each value of n from bottom (1) to top (4):
n = 1 → 2(1) + 1 = 3
n = 2 → 2(2) + 1 = 5
n = 3 → 2(3) + 1 = 7
n = 4 → 2(4) + 1 = 9
Add all results: 3 + 5 + 7 + 9 = 24
If the expression is arithmetic, you can also use Sₙ — same answer.
24
IB-style question — evaluate and interpret a sigma expression
A company's monthly revenue forms an arithmetic sequence. In month 1 the company earns $4 500. Revenue increases by $200 each month. (a)(i) Calculate the value of ∑ from n = 1 to 12 of (4300 + 200n). (a)(ii) Describe what this value represents in context.
Step by step:
Part (a)(i) — Identify the first and last terms
n = 1: 4300 + 200(1) = $4 500 (first term)
n = 12: 4300 + 200(12) = $6 700 (last term)
12 terms total → use Sₙ = n/2 (u₁ + uₙ):
S₁₂ = 12/2 × (4500 + 6700) = 6 × 11 200 = $67 200
Part (a)(ii) — Interpret the value
$67 200 is the total revenue earned by the company over all 12 months (the full year).
(a)(i) $67 200 (a)(ii) Total revenue over 12 months (the full year)
Always write the formula first — that line alone earns a method mark. Finding n from Sₙ? You will get a quadratic. Solve it and reject the negative root — n must be a positive whole number. Applications: The word total or sum signals Sₙ. The word term or value signals uₙ. Paper 2 (GDC): You can verify Sₙ on a GDC, but still write the formula and substitution — these earn method marks even if the arithmetic is wrong.