Arithmetic, geometric or Fibonacci — the terms listed out, plus closed-form results rather than counting.
This page's default seed is 3, with a step value of 5. Run as arithmetic, that produces 3, 8, 13, 18… out to a 10th term of 48 and a sum of 255 — covered in full below. Feed the identical two numbers into the other two modes and the results diverge enormously: as geometric (the same 5 now read as a ratio), the sequence runs 3, 15, 75, 375…, and by the 10th term it has already reached 5,859,375, with a sum of 7,324,218. As a Fibonacci-style sequence seeded from 3 (starting 3, 4, then each term the sum of the two before it), growth is far gentler: 3, 4, 7, 11, 18, 29, 47, 76, 123, 199 — a 10th term of just 199. Same starting number, same step, three completely different growth shapes.
Geometric sequences make that cap necessary fast. The default ratio of 5 raised to the 199th power is a number roughly 140 digits long — far beyond anything meaningful to display, and well past the point where floating-point arithmetic keeps every digit accurate. Arithmetic and Fibonacci sequences never approach this at 200 terms; only unchecked exponential growth does, which is worth remembering any time a "constant growth rate" gets quoted a long way into the future.
Arithmetic sequences describe anything adding the same fixed amount every period — a flat annual raise, or a loan under simple interest, where the interest charged never itself earns interest. Geometric sequences describe anything multiplying by a fixed factor instead, which is exactly what compound interest does every period, and why compound growth eventually dwarfs simple growth at the same headline rate — see the Investment Calculator for that same geometric growth applied to regular contributions rather than one lump sum. Fibonacci sits apart from both, defined by the two previous terms rather than a fixed step, which is why it needs its own term-by-term formula instead of either closed form above.
Each term adds a fixed amount. Starting at 3 with a difference of 5: 3, 8, 13, 18, 23… The nth term is a + (n−1)d, so the 10th is 3 + 9×5 = 48 — no counting required.
The sum has a closed form too: n/2 × (2a + (n−1)d), giving 255 for those ten terms. The trick behind it is Gauss's: pair the first term with the last, the second with the second-last, and every pair has the same total.
Each term multiplies by a fixed factor. Starting at 2 with a ratio of 3: 2, 6, 18, 54… The nth term is a · r⁽ⁿ⁻¹⁾ and the sum is a(1 − rⁿ)/(1 − r).
Geometric growth is compound interest, population growth and viral spread. It also has a property arithmetic sequences lack: when |r| < 1 the terms shrink toward zero and an infinite sum converges to a finite value, a/(1−r). Halving forever — 1 + ½ + ¼ + ⅛ … — totals exactly 2.
0, 1, 1, 2, 3, 5, 8, 13, 21… Defined by aₙ = aₙ₋₁ + aₙ₋₂ rather than by a fixed step, which makes it neither arithmetic nor geometric — though it becomes almost geometric as it grows.
The ratio between consecutive terms converges on φ ≈ 1.618, the golden ratio. 13/8 is 1.625; 89/55 is 1.6182. That convergence is why Fibonacci numbers appear in phyllotaxis and spiral packing: they are the whole-number approximation to an irrational optimum.
Arithmetic sequences add a fixed amount each step; geometric ones multiply by a fixed factor. Arithmetic grows in a straight line, geometric grows exponentially — which is why compound interest is geometric and simple interest is arithmetic.
Use the closed form. Arithmetic is a + (n−1)d, geometric is a · r^(n−1). Both jump straight to the answer, which matters when n is large.
Because the ratio of consecutive terms converges to it. As the numbers grow, dividing one by its predecessor approaches φ ≈ 1.6180339887 — the positive root of x² = x + 1, which is exactly the relationship the sequence encodes.
Yes, when the ratio is between −1 and 1. The terms shrink fast enough that an infinite sum converges to a/(1−r). Adding 1 + ½ + ¼ + ⅛ and onward forever gives exactly 2.
Compound growth is the same maths applied to money — see the Compound Interest Calculator and SIP Calculator. For the exponent arithmetic underneath, use the Exponent Calculator, and for very long sequences the Big Number Calculator keeps terms exact. Sequences look different in other bases; the base converter moves terms between decimal, binary, octal and hexadecimal.