Sequence & Series Calculator
Compute finite arithmetic and geometric terms and sums with substituted working. Supported integer inputs use exact BigInt; decimal and other paths are approximate. Check term-count and ratio limits.
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Try an example
Enter the first term, difference (ratio) and count to see the n-th term and the sum.
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Sequence & Series Calculator
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How to use
- 1
Pick the sequence type
Choose arithmetic or geometric.
- 2
Enter a₁, d (or r) and n
Type the first term, common difference or ratio, and the term count. Example buttons are available.
- 3
Read the term and the sum
The n-th term and the sum appear with the substituted formulas shown.
Features
- Arithmetic and geometric n-th terms and sums on one page
- Exact supported integer inputs; other paths are approximate
- Substituted general-term and sum formulas shown as working
- Clear range and ratio-0 errors; inputs are never transmitted
Use cases
Check sequence homework
Verify hand-worked general terms and sums together with the working.
Sanity-check totals
Quickly confirm sums like 1 + 2 + ⋯ + 100 = 5050.
Ground growth estimates
Get exact totals for fast-growing sequences such as ratio-2 growth.
Details
An arithmetic sequence has a constant difference d between terms; its general term is aₙ = a₁ + (n−1)d and its sum is Sₙ = n(a₁ + aₙ) ÷ 2. A geometric sequence has a constant ratio r; its general term is aₙ = a₁rⁿ⁻¹ and its sum, for r ≠ 1, is Sₙ = a₁(1 − rⁿ) ÷ (1 − r). The tool shows every substitution into these formulas.
BigInt is used for integer first term and difference/ratio magnitudes at most 10⁹. Geometric integer calculations require n≤500, or n≤100 for ratio magnitude above 10. Other paths use floating point, normally six decimal places or scientific notation for small or large values.
n is an integer 1–10,000. Ratio 0 is unsupported by this tool; a multiply-by-zero recurrence can still be written mathematically. Unsafe integer inputs and numeric range failures produce errors. Input calculations run in the browser.
FAQ
What is the difference between arithmetic and geometric sequences?
Arithmetic sequences add a constant difference (1, 4, 7, 10…); geometric sequences multiply by a constant ratio (1, 2, 4, 8…). Addition versus multiplication.
Why does the arithmetic sum formula look like that?
Adding the sum forward and backward pairs terms so each pair totals a₁ + aₙ, and there are n such pairs — hence Sₙ = n(a₁ + aₙ) ÷ 2. It is the classic "Gauss" trick of pairing 1 with 100.
What happens when the geometric ratio r is 1?
Every term equals the first term, so the sum is simply a₁ × n. The formula Sₙ = a₁(1 − rⁿ) ÷ (1 − r) divides by zero at r = 1, and the tool switches to a₁ × n automatically.
How large can the numbers get?
BigInt requires integer first term and difference/ratio magnitudes at most 10⁹. Geometric calculations require n≤500, or n≤100 for ratio magnitude above 10. Other paths are approximate. Integer input magnitudes above 9,007,199,254,740,991 are rejected.
Related guide
GuideHow to Use the Laws of Exponents: Product, Quotient, Power and More
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GuideHow to Do Prime Factorization
Factor 360 with a step diagram and comparison tables. Verify product and exponent forms, derive 24 positive divisors, choose GCD/LCM or fraction tools, and check the 15-digit input limit and recalculation workflow.
GuideArithmetic and Geometric Series: How to Find the Sum of a Sequence
Compare arithmetic and geometric terms and sums with worked tables. Count terms correctly, handle r=1 and negative ratios, separate finite and infinite sums, and verify integer versus decimal calculator limits.
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All processing happens in your browser. Your files are never uploaded.
Verified: Known values (5050, 205, 127, 93), the r = 1 case, ratio-0 errors, decimal inputs and the English page are covered by browser tests
Did you know?
The famous tale of young Gauss summing 1 through 100 instantly pairs 1 with 100: 101 × 100 ÷ 2 = 5050. The sum formula generalizes exactly that pairing idea.