Three Kinds of Sequence
| Type | Rule | nth term | Example |
|---|---|---|---|
| Arithmetic | Add a constant | a + (n−1)d | 3, 8, 13, 18, 23 |
| Geometric | Multiply by a constant | a × rn−1 | 3, 6, 12, 24, 48 |
| Fibonacci | Sum of the previous two | Fₙ₋₁ + Fₙ₋₂ | 0, 1, 1, 2, 3, 5, 8 |
Summing a Series
Arithmetic: S = n/2 × (first + last). The classic story has Gauss discovering this as a schoolboy asked to add 1 to 100: pairing the numbers from both ends gives fifty pairs of 101, so 5,050.
Geometric: S = a(1 − rⁿ) / (1 − r). When |r| < 1 the terms shrink and the infinite sum converges to a/(1 − r) — the reason 0.999... equals exactly 1, being the geometric series 0.9 + 0.09 + 0.009 + ...
Linear Against Exponential
Arithmetic sequences grow by a fixed amount and geometric ones by a fixed proportion. Over enough terms, any geometric sequence with ratio above 1 overtakes any arithmetic sequence, whatever the difference. This is the same asymmetry behind compound interest, population growth and the wheat-and-chessboard problem, where doubling 64 times reaches 18 quintillion grains.
Fibonacci and the Golden Ratio
The ratio of consecutive Fibonacci numbers converges to φ ≈ 1.618, the golden ratio. The sequence appears in the spiral arrangement of sunflower seeds, pine cone scales and branching patterns — not through mysticism, but because packing elements at the golden angle of 137.5° produces the most efficient use of space.
Where Sequences Appear
- Arithmetic — simple interest, straight-line depreciation, fixed monthly saving.
- Geometric — compound interest, radioactive decay, population growth, algorithmic complexity.
- Fibonacci — biological growth patterns, and as a search technique in optimisation.
Frequently Asked Questions
How do I identify a sequence type?
Take differences between consecutive terms. Constant differences mean arithmetic. If instead the ratios are constant, it is geometric. Neither, and it may be quadratic, Fibonacci-like, or something else entirely.
Can a geometric sequence have a negative ratio?
Yes — the terms then alternate in sign: 3, −6, 12, −24. The sum still converges if the absolute value of the ratio is below 1.
Why does an infinite series sometimes have a finite sum?
Because the terms shrink fast enough. Adding 1/2 + 1/4 + 1/8 + ... forever approaches exactly 1, never exceeding it. This resolves Zeno's paradox: infinitely many steps can cover a finite distance in finite time.