To find a general rule, organise your results in a table, describe the pattern, write a rule using n, test it on new cases (verify) and explain why it works (justify). For a linear pattern with a constant difference d, the rule is dn + (first term − d).
How do I tell the difference between arithmetic, polynomial and geometric sequences?
Look at how you get from one term to the next. If you add the same amount each time, the sequence is arithmetic (a linear rule). If the amount you add changes, but the differences settle down to a constant after two or three rounds, the rule is polynomial (quadratic or cubic). If you multiply by the same amount each time, the sequence is geometric, and taking differences will never give you a constant. Test the ratios instead: term 2 ÷ term 1, term 3 ÷ term 2, and so on.
| Sequence type | Test | What is constant? | Example | Rule |
|---|---|---|---|---|
| Arithmetic (linear) | Subtract consecutive terms | The first difference, Un − Un−1 | 3, 7, 11, 15, … (+4) | Un = 4n − 1 |
| Polynomial (quadratic, cubic) | Subtract consecutive terms, then repeat | The second or third difference | 2, 6, 12, 20, … (+4, +6, +8, then +2) | Un = n² + n |
| Geometric | Divide consecutive terms | The ratio, Un ÷ Un−1 | 3, 6, 12, 24, … (× 2) | Un = 3 × 2n−1 |
What happens if you take differences of a geometric sequence?
Differences of 3, 6, 12, 24, 48, … never become constant
The differences never settle to a single number like +2 or +5. Each row is just a shifted copy of the sequence itself: 3, 6, 12, 24, then 3, 6, 12. That is the giveaway: the terms are being multiplied by 2 every time, so the sequence is geometric with ratio r = 2.
Test the ratios instead
Every ratio is × 2. For a geometric sequence the rule is Un = a × rn−1, where a is the first term and r is the ratio. Here a = 3 and r = 2, so Un = 3 × 2n−1. Check: n = 4 gives 3 × 2³ = 24 ✓.
The key idea
Criterion B isn’t just about spotting a pattern. The higher levels need a general rule that works for any case, a check on new cases and a reason why the rule works.
“Add 3 each time” describes the pattern, but it isn’t a general rule. A rule such as 3n + 1 lets you find the 100th term straight away.
The differences between terms tell you what type of rule to look for. If the first differences are constant, the rule is linear (an + b). If the second differences are constant, it is quadratic (an² + bn + c). If the third differences are constant, it is cubic (an³ + bn² + cn + d).
From a pattern to a rule. Count, organise the results in a table, and look at how much is added each time. The difference tells you the number in front of n.
Here the difference is 3, so the rule starts 3n. The first shape has 4 matches, and 3 × 1 = 3, so add 1: 3n + 1.
Which type of rule is it? Keep taking differences until they are constant
The level at which the differences become constant tells you the type of rule. The constant itself tells you the number in front of the highest power: divide it by 1 for linear, by 2 for quadratic and by 6 for cubic.
First differences constant → linear
+4 every time, so the rule is 4n + 1. A linear rule looks like an + b.
Second differences constant → quadratic
The second differences are +2, and 2 ÷ 2 = 1, so the rule starts with 1n²: here it is n² + n. A quadratic rule looks like an² + bn + c.
Third differences constant → cubic
The third differences are +6, and 6 ÷ 6 = 1, so the rule starts with 1n³: here it is simply n³. A cubic rule looks like an³ + bn² + cn + d.
Worked examples
Example 1Squares are made from matchsticks in a row. 1 square uses 4 matches, 2 squares use 7 and 3 squares use 10. Find a rule.
Matches in n squares
n1234term4710131st difference+3+3+3Organise, generalise, verify, justify
- Difference = +3 each timeOrganise: the table above
- Rule starts 3nConstant difference → 3 × n
- n = 1: 3 + 1 = 4What must be added? 1
- matches = 3n + 1Generalise
- n = 5: 3 × 5 + 1 = 16 ✓Verify: draw 5 squares and count 16
- 4 to start, then +3 per squareJustify: each new square shares a side
Answer: matches = 3n + 1
Example 2Find a rule for the sequence 5, 9, 13, 17.
The sequence 5, 9, 13, 17
n1234term5913171st difference+4+4+4A linear rule: difference × n, then adjust
- Difference = +4So the rule starts 4n
- n = 1: 4 + ? = 5First term is 5
- term = 4n + 14 × 1 + 1 = 5 ✓
- n = 3: 4 × 3 + 1 = 13 ✓Verify with another term
Answer: 4n + 1
Example 3Find a rule for 2, 6, 12, 20 (rectangles of dots, n by n + 1).
Second differences constant → a quadratic rule
- 2, 6, 12, 20Differences +4, +6, +8
- 2nd difference = +2Constant, so the rule has n²
- Each shape is n × (n + 1) dotsUse the structure of the pattern
- dots = n(n + 1) = n² + nGeneralise
- n = 5: 25 + 5 = 30 ✓Verify: a 5 × 6 rectangle has 30 dots
Answer: n² + n
Example 4A cell splits into 3 every hour. There are 2 cells to start with, then 6, 18, 54, … Find a rule for the number of cells after n − 1 hours (the nth term).
Differences: +4, +12, +36, never constant
n1234term2618541st difference+4+12+36Ratios: ×3 every time
n1234term261854ratio×3×3×3A geometric rule: first term × ratio to the power n − 1
- 6 ÷ 2 = 18 ÷ 6 = 54 ÷ 18 = 3Constant ratio → geometric, r = 3
- a = 2, r = 3a is the first term
- Uₙ = a × rⁿ⁻¹Geometric rule
- Uₙ = 2 × 3ⁿ⁻¹Generalise
- n = 5: 2 × 3⁴ = 2 × 81 = 162 ✓Verify: 54 × 3 = 162
- Each cell becomes 3, so × 3 each hourJustify: multiplied, not added
Answer: Uₙ = 2 × 3ⁿ⁻¹
Common mistakes
- Stopping at a description in words (“add 3”) instead of a general rule with n.
- Testing the rule only on the cases used to find it. Verifying means trying a new case.
- Leaving out the justification: why the rule works, based on the structure of the pattern.
- Assuming the rule is linear without checking the differences. If the first differences change, take the second differences.
- Taking differences of a geometric sequence and hoping they become constant. If each row of differences looks like a shifted copy of the sequence, divide consecutive terms instead.
Practice questions
- 6, 11, 16, 21, …
Show answer
5n + 1
- 10, 8, 6, 4, …
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12 − 2n
- 1, 4, 9, 16, …
Show answer
n²
- 3, 8, 15, 24, …
Show answer
n² + 2n
- 3, 18, 57, 132, 255, … (hint: take differences three times)
Show answer
2n³ + n. First differences 15, 39, 75, 123; second 24, 36, 48; third 12, 12, so the rule starts 2n³.
- 5, 15, 45, 135, … (hint: the differences never become constant)
Show answer
Geometric: 5 × 3ⁿ⁻¹. Each term is 3 times the one before, so a = 5 and r = 3.
More topic explainers
- Solving linear equations (MYP 2–3)
- Pythagoras’ theorem (MYP 3)
- Right-angled trigonometry (SOH CAH TOA) (MYP 4)
Find more free resources, or look up command terms in the glossary.