How to use (3 steps)
- Choose the sequence type (arithmetic or geometric) and what to solve: nth term, sum, common difference/ratio, number of terms, or infinite sum.
- Enter the known values. Use the ratio method dropdown when solving a geometric common ratio.
- Results update automatically with formulas, steps, and a preview of the first few terms. Copy the URL to share the exact setup.
Arithmetic sequences use constant differences; geometric sequences use constant ratios. Keep n as a positive integer for sums.
Enter decimals or integers. Negative differences/ratios are allowed; the calculator flags unsupported cases automatically.
Sum of first n terms
The first 10 terms form a clear pattern; the sum is highlighted above.
Formula used
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Formulas & interpretation
- Arithmetic: a_n = a1 + (n−1)d and S_n = n/2 × (2a1 + (n−1)d); d = (a_n − a1)/(n−1).
- Geometric: a_n = a1 × r^(n−1); S_n = a1 × (r^n − 1)/(r − 1); if r = 1 then S_n = n × a1.
- Infinite geometric series converges only if |r| < 1; then S∞ = a1/(1 − r).
- n from geometric inputs is limited to r > 0, r ≠ 1, and a1, a_n sharing the same sign for real solutions.
All calculations run in your browser only. Copy the URL if you want to share a specific scenario.
FAQ
Which formulas does this calculator use?
Arithmetic: a_n = a1 + (n−1)d and S_n = n/2 × (2a1 + (n−1)d). Geometric: a_n = a1 × r^(n−1); S_n = a1 × (r^n−1)/(r−1) or S_n = n×a1 when r = 1; infinite series S∞ = a1/(1−r) when |r| < 1.
When does the infinite geometric series converge?
Only when the absolute value of the common ratio is less than 1 (|r| < 1). If |r| ≥ 1, the series diverges and S∞ is not shown.
What if n is not an integer?
The tool warns you when n is not an integer. For sums, n should be a positive integer term count; a non-integer result means the given a1, d/r, and a_n combination does not align with a valid whole number of terms.
Are my inputs stored anywhere?
No. Everything is calculated in your browser only. Use the “Copy URL” button if you deliberately want to share the current inputs with someone else.
Comments
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