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.
- Press Calculate to generate 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.
Arithmetic or geometric: which one should you use?
Pick the model from the pattern, not from the size of the numbers. Arithmetic sequences add the same amount each step; geometric sequences multiply by the same factor each step.
| Known pattern | Use this setting | Example |
|---|---|---|
| Each term increases or decreases by a fixed amount. | Arithmetic, solve for a_n, S_n, or d. |
3, 5, 7, 9 has common difference 2. |
| Each term is multiplied by a fixed factor. | Geometric, solve for a_n, S_n, or r. |
2, 6, 18, 54 has common ratio 3. |
| The terms shrink toward zero. | Geometric infinite sum, only when |r| < 1. |
10, 5, 2.5, ... has infinite sum 20. |
If a word problem asks for a total accumulated amount, start with S_n. If it asks for a specific position such as the 20th term, start with a_n.
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.
How do I know whether to solve for the nth term or the sum?
Use nth term when you need one position in the sequence, such as the 12th payment or 20th term. Use sum when the question asks for a total across the first n terms.
Can the common ratio be negative or fractional?
Yes. A negative ratio makes signs alternate, and a fractional ratio can make the terms shrink. The infinite sum is available only when the absolute value of the ratio is less than 1.
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