What Is a Sequence?
A sequence is an ordered list of numbers. Each number in the sequence is a term. The first term is a₁, the second is a₂, and the nth term is aₙ. Many sequences follow a rule that lets you predict any term from its position.
The three most common patterns in elementary mathematics are arithmetic, geometric, and Fibonacci.
Arithmetic Sequences
An arithmetic sequence has a constant difference between consecutive terms. That constant is called the common difference d.
d = aₙ − aₙ₋₁ (must be the same for all n)Nth-term formula:
aₙ = a₁ + (n − 1) × dExamples:
- 2, 4, 6, 8, 10 — d = 2 (even integers)
- 100, 95, 90, 85 — d = −5 (counting down by fives)
- 1, 1.5, 2, 2.5 — d = 0.5 (fractional step)
How to identify: Subtract consecutive terms. If all differences are equal, it’s arithmetic.
Worked Example
Sequence: 2, 4, 6, 8, 10 (five terms)
Detection: Differences: 4−2=2, 6−4=2, 8−6=2, 10−8=2. All equal → arithmetic with d = 2.
Extending by 3 terms:
a₆ = 10 + 2 = 12
a₇ = 12 + 2 = 14
a₈ = 14 + 2 = 16Result: Next three terms are 12, 14, 16.
General formula: aₙ = 2 + (n − 1) × 2 = 2n. Check: a₅ = 2×5 = 10. ✓
Geometric Sequences
A geometric sequence has a constant ratio between consecutive terms. That constant is called the common ratio r.
r = aₙ / aₙ₋₁ (must be the same for all n)Nth-term formula:
aₙ = a₁ × rⁿ⁻¹Examples:
- 2, 6, 18, 54 — r = 3 (tripling)
- 100, 50, 25, 12.5 — r = 0.5 (halving)
- 1, −2, 4, −8 — r = −2 (alternating sign)
How to identify: Divide consecutive terms. If all ratios are equal, it’s geometric.
Sum of a geometric series:
Sₙ = a₁ × (rⁿ − 1) / (r − 1) for r ≠ 1When |r| < 1, the infinite series converges to a₁ / (1 − r).
Fibonacci-Like Sequences
In a Fibonacci-like sequence, each term is the sum of the two preceding terms:
aₙ = aₙ₋₁ + aₙ₋₂The original Fibonacci sequence starts 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 … but any starting pair produces a Fibonacci-like sequence. For example, 2, 5, 7, 12, 19, 31, 50 is Fibonacci-like because each term is the sum of the two before it.
How to identify: Check whether each term equals the sum of the two before it. This requires at least three terms for the check to be meaningful.
Ratio property: The ratio of successive Fibonacci terms approaches the golden ratio φ = (1 + √5)/2 ≈ 1.618 as n → ∞, regardless of the starting pair.
Comparing the Three Patterns
| Property | Arithmetic | Geometric | Fibonacci |
|---|---|---|---|
| Rule | Constant difference d | Constant ratio r | aₙ = aₙ₋₁ + aₙ₋₂ |
| Growth | Linear | Exponential | Approximately exponential (φⁿ) |
| Key parameter | d = a₂ − a₁ | r = a₂ / a₁ | Starting pair (a₁, a₂) |
| Minimum terms needed to detect | 2 | 2 | 3 |
Unknown Sequences
Many sequences follow none of the three patterns above:
- Square numbers: 1, 4, 9, 16, 25 — each term is n²
- Primes: 2, 3, 5, 7, 11, 13 — no simple recursive rule
- Powers of 2: 1, 2, 4, 8, 16 — this IS geometric with r = 2
- Alternating: 1, −1, 1, −1 — geometric with r = −1
If the sequence does not match arithmetic, geometric, or Fibonacci, you may need more terms, or it may belong to a more complex family requiring calculus (polynomial, exponential, or trigonometric fits).
Practical Uses
Financial modeling: Regular deposits into a savings account form an arithmetic sequence of contributions; the balance grows as a geometric series when interest compounds.
Population growth: Modelled as geometric (constant percentage growth per period) when resources are unlimited.
Signal processing: Discrete-time signals use arithmetic indexing; the z-transform analysis involves geometric series.
Algorithm analysis: Loops that halve their input at each step run in O(log n) time — a geometric reduction in problem size.