A sequence is arithmetic when consecutive terms differ by a fixed amount (constant difference) and geometric when they are connected by a fixed multiplier (constant ratio). To classify a sequence, subtract consecutive terms to check for a constant difference, or divide them to check for a constant ratio.
What is an arithmetic sequence?
An arithmetic sequence (also called a linear sequence) has a fixed common difference, d. To move from one term to the next, you always add (or subtract) d.
Examples:
- 3, 7, 11, 15, 19 — common difference d = +4
- 20, 15, 10, 5, 0 — common difference d = −5
- 1, 1.5, 2, 2.5, 3 — common difference d = +0.5
The differences between consecutive terms are all equal. If you check: 7 − 3 = 4, 11 − 7 = 4, 15 − 11 = 4 — constant, so this is arithmetic.
What is a geometric sequence?
A geometric sequence has a fixed common ratio, r. To move from one term to the next, you always multiply by r.
Examples:
- 2, 6, 18, 54, 162 — common ratio r = 3
- 100, 50, 25, 12.5 — common ratio r = 0.5
- 1, −2, 4, −8, 16 — common ratio r = −2
The ratios between consecutive terms are all equal. Check: 6 ÷ 2 = 3, 18 ÷ 6 = 3, 54 ÷ 18 = 3 — constant, so this is geometric.
How do you identify which type a sequence is?
Use this two-step test:
- Subtract consecutive terms. If the differences are all equal → arithmetic.
- Divide consecutive terms. If the ratios are all equal → geometric.
- If neither gives a constant value → neither (could be quadratic, Fibonacci, or another type).
Example: classify 5, 10, 20, 40.
- Differences: 10 − 5 = 5, 20 − 10 = 10, 40 − 20 = 20. Not constant → not arithmetic.
- Ratios: 10 ÷ 5 = 2, 20 ÷ 10 = 2, 40 ÷ 20 = 2. Constant → geometric with r = 2.
Example: classify 2, 5, 8, 11, 14.
- Differences: 5 − 2 = 3, 8 − 5 = 3, 11 − 8 = 3, 14 − 11 = 3. Constant → arithmetic with d = 3. No need to check ratios.
Comparison table: arithmetic vs geometric
| Feature | Arithmetic | Geometric |
|---|---|---|
| Rule | Add a constant difference, d | Multiply by a constant ratio, r |
| Test | Differences are all equal | Ratios (term ÷ previous term) are all equal |
| nth term formula | a + (n − 1)d | a × r^(n−1) |
| Growth pattern | Straight-line (linear) | Curved (exponential) |
| Example | 3, 7, 11, 15 (d = 4) | 3, 6, 12, 24 (r = 2) |
| Decreasing example | 10, 7, 4, 1 (d = −3) | 80, 40, 20, 10 (r = 0.5) |
What other sequence types might appear at KS3?
Some sequences are neither arithmetic nor geometric:
- Quadratic sequences: the second differences (differences of differences) are constant. Example: 1, 4, 9, 16, 25 (perfect squares).
- Fibonacci-type: each term is the sum of the two before it. Example: 1, 1, 2, 3, 5, 8, 13.
- Other patterns: triangular numbers (1, 3, 6, 10, 15 — add 1, 2, 3, 4, …), cube numbers, etc.
If both the difference test and ratio test fail, look for a second-difference pattern or a "add the previous two" rule.
What mistakes do students commonly make?
- Testing just one pair of terms. A single difference or ratio being constant is not enough — check at least three consecutive pairs. If the fourth term breaks the pattern, the sequence is not the type you thought.
- Confusing arithmetic and geometric. "Arithmetic adds, geometric multiplies." Write this rule on your formula sheet and glance at it during a test.
- Using the wrong formula. The nth term of an arithmetic sequence uses addition; the nth term of a geometric sequence uses exponentiation. Swapping them gives completely wrong answers.
Frequently asked questions
Can a sequence be both arithmetic and geometric?
Only in the trivial case where all terms are equal (e.g. 5, 5, 5, 5, …). Here d = 0 and r = 1. Any non-constant sequence can only be one or the other, not both.
What does a negative common ratio look like?
It makes the sequence alternate between positive and negative values. For example, 2, −6, 18, −54 has r = −3. The terms grow in magnitude but flip sign each time. The ratio test still works: (−6) ÷ 2 = −3, 18 ÷ (−6) = −3, (−54) ÷ 18 = −3 — constant, so it is geometric.
What if the common ratio is between 0 and 1?
The sequence decreases towards zero but never reaches it. For example, 64, 32, 16, 8, 4, 2 has r = ½. Each term is half the previous one. This is still geometric — the defining feature is the constant ratio, not whether the sequence grows or shrinks.
How are sequence types linked to their graphs?
If you plot the term number (n) on the x-axis and the term value on the y-axis, an arithmetic sequence gives points on a straight line (linear graph) and a geometric sequence gives points on a curve (exponential shape). This visual check can confirm your classification quickly.
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