Sequence Generator — Arithmetic, Geometric, Fibonacci & Custom Step
Generate number sequences in four modes — Arithmetic (constant difference), Geometric (constant ratio), Fibonacci (sum of two preceding terms) and Custom step (any repeating step pattern). Furthermore, an SVG chart draws the sequence shape instantly so you can see whether it is linear, exponential or oscillating at a glance. Statistics including min, max, mean and last term appear automatically. Download any sequence as CSV.
How to use the Sequence Generator
Select the sequence type
Click one of the four tabs — Arithmetic, Geometric, Fibonacci or Custom. Furthermore, each tab shows the relevant parameters. Arithmetic needs a start value and a constant step. Geometric needs a start value and a multiplying ratio. Fibonacci accepts custom first and second terms.
Set the parameters and term count
Enter the start value, step or ratio, and the number of terms to generate. Furthermore, the term count controls how many values appear in the output. For Geometric sequences, set the decimal places to control rounding precision. Custom mode lets you enter a repeating pattern of step values.
Click Generate and read the output
Click Generate to produce the sequence. Furthermore, all terms appear comma-separated in the output box. The SVG chart below draws the shape of the sequence instantly — showing whether values grow, shrink or follow a non-linear curve.
Review the statistics strip
The stats strip below the output shows the term count, minimum value, maximum value, mean and the last term. Furthermore, these figures are useful for understanding the range and central tendency of the sequence without manual calculation. The last term is particularly useful for Fibonacci sequences where manual calculation is error-prone.
Copy or download
Click Copy to copy the comma-separated sequence to your clipboard. Furthermore, click CSV to download the sequence as a spreadsheet-compatible CSV file. This is useful for importing sequence data into Excel, Python, R or any other analytical tool.
The four sequence types compared
Each sequence type has a different mathematical structure and produces a distinctly different chart shape. Furthermore, recognising these shapes helps identify sequence types in data and understand their growth behaviour.
| Type | Formula for term n | Chart shape | Growth |
|---|---|---|---|
| Arithmetic | a + (n-1) × d | Straight line | Linear — grows by the same amount each step |
| Geometric | a × r^(n-1) | Exponential curve | Exponential — grows by the same factor each step |
| Fibonacci | F(n-1) + F(n-2) | Exponential-like curve | Approximately geometric with ratio ≈ 1.618 (golden ratio) |
| Custom | Previous + pattern[n % length] | Varies by pattern | Depends on step pattern — can be irregular |
The custom step pattern
Custom mode generates a sequence where the step between consecutive terms cycles through a repeating pattern. Furthermore, a pattern of "1,2,3" produces steps of 1, then 2, then 3, then 1 again — creating an irregular but structured sequence. This models real-world patterns like staggered schedules, rhythmic structures and non-uniform sampling intervals. Moreover, it is useful for creating test data with a specific structural pattern.
Key formulas for number sequences
Each sequence type has a direct formula for calculating any term without generating all preceding terms. Furthermore, understanding these formulas helps predict sequence behaviour and verify output.
d = common difference (arithmetic)
r = common ratio (geometric)
n = term position (1-indexed)
Fibonacci = F(n) = F(n-1) + F(n-2), where F(1)=0, F(2)=1
Sum formulas
The sum of an arithmetic sequence of n terms is n × (first term + last term) ÷ 2. Furthermore, this is the formula attributed to Gauss — who reputedly calculated the sum 1 to 100 as a child by pairing each number with its complement. For a geometric sequence, the sum is a₁ × (1 − rⁿ) ÷ (1 − r) for r ≠ 1. Moreover, these formulas allow instant sum calculation without adding every term individually.
Worked example: arithmetic sequence for a savings plan
A saver deposits £200 in month 1 and increases their deposit by £25 every month for 12 months. The arithmetic sequence of monthly deposits:
| Setting | Value |
|---|---|
| Start (a₁) | 200 |
| Step (d) | 25 |
| Terms | 12 |
Output: 200, 225, 250, 275, 300, 325, 350, 375, 400, 425, 450, 475
What is a number sequence?
A number sequence is an ordered list of numbers following a defined rule. Furthermore, each term in the sequence relates to the previous term through a mathematical operation — addition (arithmetic), multiplication (geometric) or addition of preceding terms (Fibonacci). Sequences appear throughout mathematics, science, finance and computing. Moreover, understanding sequence types helps recognise patterns in data and apply appropriate analytical models.
Arithmetic sequences describe constant-rate growth — a salary increasing by the same amount each year, a vehicle travelling at constant speed. Furthermore, geometric sequences describe proportional growth — compound interest, bacterial doubling, radioactive decay. Fibonacci sequences appear in nature — the spiral arrangement of seeds in sunflowers, the branching of trees and the structure of shells all approximate Fibonacci proportions. Moreover, the Fibonacci sequence converges to the golden ratio (φ ≈ 1.618) as terms increase.
Sequences in computer science
Computer science uses sequences extensively — from generating test data to implementing algorithms. Furthermore, arithmetic sequences produce evenly spaced sample indices for signal processing. Geometric sequences model algorithm complexity — O(2ⁿ) exponential algorithms follow a geometric sequence of operation counts. Moreover, Fibonacci sequences appear in algorithm design — the Fibonacci search technique and heap data structures both exploit Fibonacci properties.
Why the chart preview matters
A chart immediately reveals the growth behaviour of a sequence that numbers alone obscure. Furthermore, a geometric sequence with ratio 1.1 looks like slow linear growth for the first ten terms — but the chart shows the curve bending upward. Spotting exponential behaviour early is critical for planning — a cost or metric growing geometrically will accelerate dramatically. Moreover, the flat line of arithmetic sequences versus the steep curve of geometric sequences makes the difference viscerally obvious in a way that a list of numbers does not.
Educators use sequence generators to create practice problems. Furthermore, students learning arithmetic and geometric sequences need large numbers of examples to practice with. Generating fifty different sequences with varying parameters provides varied practice material in seconds. Moreover, the chart preview gives students immediate visual feedback on whether their chosen parameters produce the expected shape — linear, exponential, or Fibonacci-like.
Custom sequences for scheduling
Custom step patterns are useful for generating irregular but structured schedules. Furthermore, a maintenance schedule might follow a pattern of 30, 60, 90, 30 days — repeating this cycle over several years. The custom sequence generator converts any repeating step pattern into a full schedule instantly. Moreover, the CSV export allows the generated schedule dates to be imported directly into a calendar or project management tool.
Frequently asked questions
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