Spot arithmetic patterns fast
Big idea: An arithmetic sequence changes by the same amount each time.
That fixed amount is called the common difference.
| Sequence | What changes? | Arithmetic? |
|---|---|---|
| 4, 7, 10, 13, ... | +3 each time | Yes |
| 15, 12, 9, 6, ... | -3 each time | Yes |
| 2, 4, 8, 16, ... | ×2 each time | No |
| 5, 5.5, 6, 6.5, ... | +0.5 each time | Yes |
Quick checks
- Add or subtract the same number each time -> arithmetic.
- Multiply or divide by the same number -> not arithmetic.
- The common difference can be negative or decimal.
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Find the common difference
To find the common difference, subtract one term from the next term.
Easy examples:
- 8, 12, 16, 20, ... -> d = 12 - 8 = 4
- 21, 18, 15, 12, ... -> d = 18 - 21 = -3
- 2.5, 3.0, 3.5, 4.0, ... -> d = 0.5
Common mistake
- Looking at only the first and last term
- Using multiplication instead of subtraction
- Forgetting the negative sign
Correct method
- Compare two neighbouring terms
- Subtract in the same order each time
- Keep the sign on the answer
IB-style question — common difference from a table [2 marks]
A taxi company charges by distance.
The table shows the fare for the first three distances: 1 km → $4.50, 2 km → $6.20, 3 km → $7.90.
Show that the fares form an arithmetic sequence, and write down the common difference.
Step by step
- Subtract each fare from the next one.
- The difference is the same each time, so the sequence is arithmetic.
Final answer
Common difference d = $1.70, constant — so the fares are arithmetic.
Exam tips
- Use second term minus first term.
- If the sequence is going down, the common difference is negative.
- Decimals are allowed.
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Answer exam-style questions and get AI feedback that shows you exactly what examiners want to see in a full-marks response.
Use the nth term rule
Rule: For an arithmetic sequence: uₙ = u₁ + (n − 1)d
Here, u₁ is the first term and d is the common difference.
Worked example:
- Sequence: 5, 8, 11, 14, ...
- u₁ = 5 and d = 3
- uₙ = 5 + (n − 1)3
- So uₙ = 3n + 2
Worked example — Library shelves
A library puts books on shelves in an arithmetic pattern.
Shelf 1 has 12 books. Shelf 2 has 17 books. Shelf 3 has 22 books, and so on.
Find the number of books on shelf 10.
Step by step
- Step 1 — Identify u₁ and d.
u₁ = 12 (shelf 1).
d = 17 − 12 = 5 (the fixed number of extra books each shelf). - Step 2 — Apply uₙ = u₁ + (n − 1)d with n = 10.
- Step 3 — Evaluate.
Final answer
There are 57 books on shelf 10.
Worked example — Theatre row prices
Tickets for a theatre are priced in an arithmetic pattern.
Row 1 costs $8. Row 2 costs $11. Row 3 costs $14, and so on.
Find the price of a ticket in row 20.
Step by step
- Step 1 — Identify u₁ and d.
u₁ = $8 (the price in row 1).
d = 11 − 8 = $3 (the fixed increase per row). - Step 2 — Apply uₙ = u₁ + (n − 1)d with n = 20.
- Step 3 — Evaluate.
Final answer
A ticket in row 20 costs $65.
Exam tips
- Write down u₁ and d first.
- Use brackets carefully.
- Then simplify.