Number Sequence Calculator
Generate arithmetic, geometric or Fibonacci-style sequences, with the nth term and running sum.
How to use this calculator
- 1Choose a sequence type and enter its starting value(s).
- 2Set how many terms to generate — up to 500.
How the calculation works
Arithmetic: aₙ = a₁ + (n−1)d Geometric: aₙ = a₁ × r⁽ⁿ⁻¹⁾ Fibonacci-style: aₙ = aₙ₋₁ + aₙ₋₂- a₁
- First term
- d
- Common difference (arithmetic)
- r
- Common ratio (geometric)
- n
- Term position
The classic Fibonacci sequence is just this pattern starting at 0 and 1 — entering any other starting pair produces a "Fibonacci-like" sequence with the same defining recurrence but different values (a Lucas sequence, for example, starts at 2 and 1).
Worked example
Arithmetic: first term 2, common difference 3, 10 terms
- 1.Sequence: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29.
- 2.Sum: 10/2 × (2 + 29) = 5 × 31 = 155.
Result: 2, 5, 8, …, 29 — sum 155
Three ways a sequence can grow
An arithmetic sequence adds the same fixed amount at every step — 2, 5, 8, 11 — so it grows in a straight line. A geometric sequence instead multiplies by the same fixed amount each step — 2, 6, 18, 54 — which means it grows (or shrinks) exponentially rather than steadily. A Fibonacci-style sequence follows a different rule entirely: each term is the sum of the two before it, which produces growth that settles into its own exponential pattern over time, just built from addition rather than a fixed multiplier.
Where each type shows up
The three patterns are not just abstract exercises — each one models a distinct kind of real-world growth.
- Arithmetic sequences — simple interest, a fixed hourly wage accumulating over time, or evenly spaced measurements.
- Geometric sequences — compound interest, population growth under ideal conditions, and radioactive decay — a geometric sequence with a ratio between 0 and 1.
- Fibonacci-style sequences — the spiral arrangement of seeds in a sunflower head, the branching patterns of some plants, and idealised models of population growth.
The golden ratio hides inside the Fibonacci sequence
Divide any Fibonacci number by the one before it, and as the sequence continues, that ratio settles closer and closer to approximately 1.618 — the golden ratio. The sequence is named after Leonardo of Pisa, known as Fibonacci, who introduced it to European readers in his 1202 book Liber Abaci, though the same pattern had already appeared earlier in Indian mathematics.
What this assumes, and where it stops
Assumptions
- The pattern (common difference, ratio, or the previous-two-terms rule) stays exactly the same across every term.
Limitations
- Very large term counts with a geometric ratio greater than 1 can produce numbers too large to display precisely — the sum is shown in scientific notation where needed.
Common questions
What's the difference between arithmetic and geometric sequences?
An arithmetic sequence adds the same amount each step (2, 5, 8, 11 — always +3). A geometric sequence multiplies by the same amount each step (2, 6, 18, 54 — always ×3). The distinction matters because geometric sequences grow (or shrink) exponentially, while arithmetic sequences grow linearly.
Why does the geometric sum formula not work when the ratio is 1?
The formula divides by (r − 1), which is zero when r = 1 — every term is identical in that case, so the sum is simply the first term multiplied by the number of terms instead, which is what this calculator switches to automatically.
Formula and content last reviewed on .
Results are estimates for information only, not professional advice.
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