Calculator
Example Data Table
| Example | Inputs | Outputs |
|---|---|---|
| Growth sequence | a₁=3, r=2, n=6 | a₆=96, S₆=189, terms: 3, 6, 12, 24, 48, 96 |
| Decay sequence | a₁=80, r=0.5, n=5 | a₅=5, S₅=155 |
| Solve ratio | a₁=5, k=4, a_k=40 | r=2, check: a₄=40 |
Formula Used
- Nth term aₙ = a₁ · r^(n−1)
- Sum of first n Sₙ = a₁(1 − r^n)/(1 − r) for r ≠ 1
- Sum when r = 1 Sₙ = a₁ · n
- Partial sum S(m..n) = Sₙ − S(m−1)
- Solve n n = 1 + log(aₙ/a₁)/log(r) (real solution requires r>0 and aₙ/a₁>0)
- Solve r r = (a_k/a₁)^(1/(k−1)) (real solution requires a_k/a₁ ≥ 0)
How to Use This Calculator
- Select a calculation mode that matches your goal.
- Enter a₁, r, and indexes (n or m..n) as needed.
- Press Submit to display results above the form.
- Use Download CSV or Download PDF to export the latest results.
- Reset to clear inputs and start a new calculation.
FAQs
1) What is a geometric sequence?
A sequence where each term is the previous term multiplied by a constant ratio r. Examples include compounding growth and repeated halving.
2) What happens if r equals 1?
All terms become equal to a₁. The sum of the first n terms becomes Sₙ = a₁·n, and solving n from a target aₙ is not unique.
3) Can I use negative ratios?
Yes for generating terms, nth term, and sums. However, solving n using logarithms needs r>0 and aₙ/a₁>0 to keep solutions real-valued.
4) Why does the sequence preview show fewer rows?
The on-page preview is trimmed for readability. The CSV export includes all computed terms for the selected mode and range.
5) What if my inputs are decimals?
Decimals work for a₁ and r. Index inputs like n, m, and k are treated as integers because term positions must be whole numbers.
6) How is partial sum computed?
The calculator uses S(m..n) = Sₙ − S(m−1). This avoids adding each term manually and remains accurate for large n.
7) Is the PDF export accurate without libraries?
Yes for text-based summaries and a limited term list. It creates a lightweight PDF file directly, suitable for printing or saving calculations.