Sequence & Series Calculator
Get the n-th term and the sum of arithmetic or geometric sequences from the first term, difference (ratio) and count — computed exactly with BigInt. The substituted general-term and sum formulas are shown, and huge geometric results stay error-free.
Try an example
Enter the first term, difference (ratio) and count to see the n-th term and the sum.
Done
Sequence & Series Calculator
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 BigInt arithmetic for integer inputs (geometric results up to about 1000 digits)
- 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.
Integer inputs are computed with BigInt (arbitrary-precision integers): besides classics like 1 + 2 + ⋯ + 100 = 5050, geometric sums with about 31 digits (ratio 2, 100 terms) come out exactly. Inputs containing decimals are computed with floating point and shown rounded to six decimal places.
Conditions: n is an integer from 1 to 10000, and the ratio of a geometric sequence cannot be 0 (a "sequence" with ratio 0 collapses to zeros and is normally excluded). Very large geometric results hit a size limit, which the tool reports. Inputs are never transmitted; all computation happens locally in your 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?
Integer inputs are computed with BigInt, so there is no rounding error. Geometric digit counts explode with the ratio and term count — about 500 terms with |ratio| ≤ 10 is a practical ceiling, and the tool reports the limit beyond that.
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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.