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.
Free preview
This is the free notes preview
You're reading the free notes. Aimnova Pro unlocks the full study experience — and you can try it free for 7 days:
- FlashcardsLock in vocabulary and key terms with spaced repetition.
- Practice questionsAnswer exam-style questions and get instant AI marking.
- Mock exams & past-paper vaultSit full mocks and see exactly how examiners award marks.
- Personalised study planA daily plan built around your exam date and weak areas.
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.
Practice with real exam questions
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.