Number Sequence Calculator
Pick a sequence type — arithmetic, geometric, or Fibonacci — and get the nth term, the sum of the first n terms, and the sequence itself. Fibonacci values are computed exactly, even past the usual calculator limits.
Example: with Sequence type Arithmetic (add d each step) · First term (a₁) 2 · Common difference (d) 5 · Number of terms (n) 20 → nth term: 97.
- Sum of first n terms990
- Formula usedaₙ = a₁ + (n−1)d
Computed by the calculator below using its default values. Change any input to see your own numbers.
Sequence
Three classic sequences
An arithmetic sequence adds the same difference each step (2, 7, 12, 17, …), so its nth term is a₁ + (n − 1)d and its sum is n(a₁ + aₙ) ÷ 2 — the average of the first and last terms times how many there are. A geometric sequence multiplies by the same ratio each step (3, 6, 12, 24, …); it grows or decays exponentially, which is why the same math prices compound interest. The Fibonacci sequence adds the two previous terms (1, 1, 2, 3, 5, 8, …) and its ratio of consecutive terms approaches the golden ratio φ ≈ 1.618.
How it’s calculated
Arithmetic: aₙ = a₁ + (n − 1)d; Sₙ = n(a₁ + aₙ) ÷ 2. Geometric: aₙ = a₁rⁿ⁻¹; Sₙ = a₁(1 − rⁿ) ÷ (1 − r) for r ≠ 1. Fibonacci: F₁ = F₂ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂, computed with exact integer arithmetic (BigInt); the sum of the first n terms equals Fₙ₊₂ − 1.
Results update as you type and are for education, not professional advice — double-check any number that matters.
Common mistakes
- Using n instead of n − 1 in the nth-term formulas — the first term already "uses up" one position.
- Applying the geometric sum formula with r = 1 — the sum is simply n × a₁ in that case.
- Assuming Fibonacci starts at 0 everywhere — conventions differ; this tool uses F₁ = F₂ = 1 and says so.
Frequently asked questions
What is the difference between arithmetic and geometric growth?
Arithmetic adds a fixed amount each step, so the plot is a straight line. Geometric multiplies by a fixed ratio, so it curves — doubling, halving, compounding. The same $100 growing by $10/year is arithmetic; growing 10%/year is geometric.
What happens to a geometric series when |r| < 1?
Terms shrink toward zero and the running sum approaches a₁ ÷ (1 − r). With a₁ = 3 and r = 0.5 the infinite sum is 6 — the calculator shows the finite sum for your n.
How large a Fibonacci number can this compute?
Up to n = 1000 exactly, using arbitrary-precision integers. F₁₀₀ is already 354,224,848,179,261,915,075 — 21 digits — far past ordinary floating-point precision.
Does the Fibonacci sequence here start with 0 or 1?
With 1, 1, 2, 3, … (F₁ = F₂ = 1). Some texts define F₀ = 0 first; that just shifts the index by one, so their F(n) is our term n.
Can the common difference or ratio be negative?
Yes. A negative d makes an arithmetic sequence decrease; a negative r makes a geometric sequence alternate signs (3, −6, 12, −24, …). The formulas handle both.