Number Sequence Calculator
Arithmetic, geometric and Fibonacci — all on one page. Enter the first term(s) and how many terms you want, and see the nth value, the running sum, the full term list and a small plot of how the sequence grows.
1. Arithmetic Sequence Calculator
Definition: aₙ = a₁ + d·(n − 1)
Example: 1, 3, 5, 7, 9, 11, 13, …
2. Geometric Sequence Calculator
Definition: aₙ = a · r^(n − 1)
Example: 1, 2, 4, 8, 16, 32, 64, 128, …
3. Fibonacci Sequence Calculator
Definition: F₀ = 0, F₁ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂
Example: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …
What is a number sequence?
A sequence is an ordered list of numbers that follows a rule. Each entry is a term, and the position of a term is its index, usually written with a subscript: a₁ is the first term, a₂ the second, and so on.
A rule can be given as a formula in n (like aₙ = 3n − 1) or as a recurrence that says how each term depends on the previous ones (like Fibonacci's Fₙ = Fₙ₋₁ + Fₙ₋₂). This page covers the three sequence families you meet most often at school and in day-to-day problems.
Sequences, one type at a time
Each card has a plain-English definition, the formulas for the nth term and running sum (with every symbol spelled out), a small growth diagram, and a worked example.
Arithmetic sequence — constant difference
An arithmetic sequence adds the same number every step. That constant is the common difference d. Because each jump is identical in size, the terms lie on a straight line — growth (or decline, if d is negative) is linear.
Geometric sequence — constant ratio
A geometric sequence multiplies by the same number every step. That constant is the common ratio r. Because each term is scaled by a fixed factor, growth is exponential — or, when |r| < 1, an exponential decay towards zero. The infinite sum only converges when |r| < 1.
Fibonacci sequence — sum of the two before
Fibonacci is defined by the simplest possible recurrence: each term is the sum of the two before it. Consecutive ratios Fₙ/Fₙ₋₁ approach the golden ratio φ ≈ 1.618…, which is why Fibonacci grows roughly like φⁿ — much slower than doubling, but still exponential. The sequence appears in phyllotaxis, honeybee family trees and combinatorial counts.
Features of this calculator
- Three distinct sub-tools on one page — no need to switch calculators for arithmetic, geometric or Fibonacci
- Renders the actual list of terms, not just the nth value
- Auto-scaled line plot for every sub-tool so linear, exponential and Fibonacci-style growth are visible at a glance
- Geometric mode switches to a log-scale y-axis when the terms span many orders of magnitude
- Fibonacci computed with BigInt and fast doubling — accurate for n in the thousands, not capped at ~F₇₈ like floating point
- Handles negative common difference / ratio, including alternating signs
- Step-by-step working using the standard closed-form formulas
Frequently asked questions
+What is the difference between a sequence and a series?
A sequence is the ordered list itself (2, 5, 8, 11, …). A series is what you get when you add the terms up (2 + 5 + 8 + 11 + …). This calculator always reports both: the list and the running total Sₙ.
+Why do Fibonacci numbers grow so fast?
Each new term adds the previous one, so once you're past the first few values the sequence is roughly multiplying by the golden ratio φ ≈ 1.618 at every step. That is exponential growth — F₃₀ is already 832 040, and F₁₀₀ has 21 digits.
+Can the common difference or ratio be negative?
Yes. A negative common difference makes an arithmetic sequence decrease (10, 7, 4, 1, −2, …). A negative common ratio makes a geometric sequence alternate signs (3, −6, 12, −24, …). Both are handled here, including in the plot.
+Do sequences start at n = 0 or n = 1?
Both conventions are common. Arithmetic and geometric tools here start at a₁. The Fibonacci tool uses the standard F₀ = 0, F₁ = 1 so the indexing lines up with what most textbooks and OEIS use.