Free Number Sequence Calculator
Calculate the nth term, sum of n terms, and generate the full sequence for arithmetic and geometric progressions. Find missing terms and identify the sequence type. Free, private — all calculations run in your browser.
Number Sequence Calculator — Arithmetic, Fibonacci & More
Generate number sequences including arithmetic progressions, geometric series, Fibonacci, prime numbers, perfect squares, triangular numbers, and factorial sequences. Shows the nth term formula and sum. Used in mathematics education, coding interviews and data analysis.
How to Use This Calculator
- 1
Select the sequence type from the dropdown.
- 2
For arithmetic/geometric sequences, set the first term and common difference/ratio.
- 3
Set how many terms to generate (up to 50).
- 4
The full sequence, sum, and nth-term formula appear instantly.
- 5
Copy the sequence or export a PDF report.
📐 How This Is Calculated
Arithmetic: a(n) = a₁ + (n−1)d | Geometric: a(n) = a₁ × rⁿ⁻¹ | Fibonacci: F(n) = F(n−1) + F(n−2)
a₁—First term of the sequenced—Common difference (arithmetic) — added between each termr—Common ratio (geometric) — multiplied between each termn—Term number (1-based index)Reference: Carl Friedrich Gauss — arithmetic series; Leonardo of Pisa "Fibonacci" (1202)
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Disclaimer
Results are provided for informational and educational purposes only. They should not be used as a substitute for professional financial, engineering, medical, or legal advice. Always verify outputs with a qualified professional before making important decisions. Roughtools makes no warranties regarding accuracy or completeness for your specific situation.
About This Number Sequence Calculator
The Number Sequence Calculator handles the two most fundamental types of mathematical sequences: arithmetic progressions (constant difference between terms) and geometric progressions (constant ratio between terms). Enter the first term and either the common difference or ratio, specify how many terms you need, and the calculator returns every term along with the running sum.
The Formulas
Real-World Applications
Arithmetic sequences model steady linear growth: annual salary increases, equal instalments, uniform acceleration in physics. Geometric sequences model exponential growth and decay: compound interest, population growth, radioactive decay, viral spread. The sum formula for geometric sequences is the mathematical foundation of annuity calculations in finance — the present or future value of a series of equal payments made at regular intervals.
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When to Use This Calculator
Find the nth term, sum of n terms, or missing terms in arithmetic and geometric progressions for coursework and examination problems.
Model savings with regular contributions (arithmetic sums) or compound growth (geometric sequences) to understand long-term portfolio value.
Calculate radioactive decay sequences, spring oscillation amplitude decreases, or any process described by a constant ratio between consecutive values.
Understand the geometric sequence of frequencies across octaves: each octave doubles frequency, creating a geometric progression with r = 2.
Identify the type and formula of a numerical sequence when developing algorithms, data patterns, or solving number puzzles.
💡 Pro Tips
The sum formula for an arithmetic sequence Sₙ = n(a₁ + aₙ)/2 has a beautiful intuition: pair the first and last terms (which sum to the same value as the second and second-to-last, and so on), and multiply by the number of pairs. Gauss reportedly used this insight as a child to sum 1 to 100 instantly: S₁₀₀ = 100 × 101 / 2 = 5,050.
For geometric sequences with ratio r = 1 + growth rate, the sum formula Sₙ = a₁(rⁿ − 1)/(r − 1) is the foundation of compound interest: the total value of regular deposits in an account with compound interest. This is why starting to invest early is so powerful — you are adding terms to a geometric sum where each term is larger than the last.
To check if a sequence is arithmetic or geometric without a calculator: divide consecutive terms (if the ratios are all equal → geometric) and subtract consecutive terms (if the differences are all equal → arithmetic). If a sequence looks like it grows faster and faster, it is probably geometric. Linear growth suggests arithmetic.
The nth term of the Fibonacci sequence F(n) = [φⁿ − (1−φ)ⁿ] / √5, where φ = (1 + √5)/2 ≈ 1.618 (the golden ratio). Despite involving irrational numbers, this formula always gives exact integers. The ratio of consecutive Fibonacci numbers converges to φ as n → ∞, which is why the Fibonacci sequence is often called a pseudo-geometric sequence.
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