Education
Series Calculator
Pick arithmetic or geometric, enter the first term, the common difference or ratio, and how many terms. The calculator returns the nth term, the sum of the first n terms, the closed-form formula used, and a preview of the early terms.
Sequence type
What to compute
e.g. 2
e.g. 3
A positive whole number. · e.g. 10
Arithmetic a_n
29
Arithmetic sequences add the same number each step; geometric sequences multiply by the same ratio each step. Both have closed formulas for the nth term and the sum of the first n terms.
Examples
Arithmetic a₁=2, d=3, n=10
a_10 = 29, S_10 = 155
Arithmetic a₁=1, d=1, n=100
a_100 = 100, S_100 = 5,050
Geometric a₁=3, r=2, n=5
a_5 = 48, S_5 = 93
Geometric a₁=1, r=0.5, n=10
a_10 ≈ 0.00195, S_10 ≈ 1.998
How it works
The calculator uses the standard closed-form formulas. They are exact for any n; no looping is needed.
Arithmetic · a_n = a₁ + (n − 1) × d · S_n = n × (a₁ + a_n) / 2
Geometric · a_n = a₁ × r^(n − 1) · S_n = a₁ × (1 − rⁿ) / (1 − r)
Related math fundamentals calculators
- Exponent calculator for raising the common ratio to any power directly.
- Scientific calculator for arithmetic checks of the closed-form result.
- Log calculator for finding n in a geometric sequence given a target a_n.
- Percentage increase calculator for converting a geometric ratio into a percent change per step.
- All education calculators.
Frequently asked questions
An arithmetic sequence adds the same number d each step: 2, 5, 8, 11, … (d = 3). A geometric sequence multiplies by the same number r each step: 3, 6, 12, 24, … (r = 2). The same calculator handles both.
a_n is the nth term of the sequence. For arithmetic, a_n = a₁ + (n − 1) × d. For geometric, a_n = a₁ × r^(n − 1). The calculator shows the formula with your actual values plugged in.
For arithmetic, S_n = n × (a₁ + a_n) / 2, the average of the first and last term times the count. For geometric with r ≠ 1, S_n = a₁ × (1 − rⁿ) / (1 − r). When r = 1, every term is a₁, so S_n is just n × a₁.
Not in this version. For |r| < 1, the infinite sum converges to a₁ / (1 − r). For |r| ≥ 1, it diverges. Choose a large n in this calculator to see the partial sum approaching the limit.
The preview shows the first 8 terms (or fewer if n < 8) for readability. The closed-form result is exact for any n; the preview is just a check.
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