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.
How to use this calculator effectively
This guide helps you use Arithmetic & Geometric Sequence Calculator in a repeatable way: define a baseline, change one variable at a time, and interpret outputs with explicit assumptions before you share or act on results.
How it works
The page applies deterministic logic to your inputs and shows rounded output for readability. Treat it as a comparison workflow: run one baseline case, adjust a single parameter, and measure both absolute and percentage deltas. If a result seems off, verify units, time basis, and sign conventions before drawing conclusions. This approach keeps your analysis reproducible across teammates and sessions.
When to use
Use this page when you need a fast estimate, a classroom check, or a practical what-if comparison. It works best for planning and prioritization steps where you need direction and magnitude quickly before investing in deeper modeling, manual spreadsheets, or formal external review.
Common mistakes to avoid
- Changing multiple parameters at once, which hides the true cause of output movement.
- Mixing units (percent vs decimal, monthly vs yearly, gross vs net) across scenarios.
- Comparing with another tool without aligning defaults, constants, and rounding rules.
- Using rounded display values as exact downstream inputs without re-checking precision.
Interpretation and worked example
Run a baseline scenario and keep that result visible. Next, modify one assumption to reflect your realistic alternative and compare direction plus size of change. If the direction matches your domain expectation and the size is plausible, your setup is usually coherent. If not, check hidden defaults, boundary conditions, and interpretation notes before deciding which scenario to adopt.
See also
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.
What should I do first on this page?
Start with the minimum required inputs or the first action shown near the primary button. Keep optional settings at defaults for a baseline run, then change one setting at a time so you can explain what caused each output change.
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