Result Summary
| Series | Inputs | n (real) | Nearest n | Sum at nearest n | Error (Sum−S) |
|---|---|---|---|---|---|
| natural | S=55 | 10.000000 | 10 | 55.000000 | 0.000000 |
Formulas Used
- Natural numbers:
S = n(n+1)/2, son = (-1 + √(1+8S)) / 2. - Arithmetic progression:
S = n/2 · [2a + (n−1)d]. Solve the quadratic inn. - Geometric progression:
S = a(1 − r^n)/(1 − r)forr ≠ 1. Closed-formn = log(1 − S(1−r)/a)/log(r)when valid; otherwise numeric.
Partial Sums Plot
The chart shows partial sums Sk versus term count k, with your target S highlighted.
Scenario Overlays
| # | Label | Series | Params | Actions |
|---|
Overlays persist in your browser (local storage). They render together on the chart.
Example Data Table
| Series | Inputs | n (real) | Nearest n | Sum at nearest n | Error (Sum−S) |
|---|---|---|---|---|---|
| Natural | S=55 | 10.000000 | 10 | 55.000000 | 0.000000 |
| Arithmetic | S=100, a=3, d=2 | 9.049876 | 9 | 99.000000 | -1.000000 |
| Geometric | S=100, a=5, r=1.2 | 8.827469 | 9 | 103.994509 | 3.994509 |
FAQs
You provide a total sum, and the tool solves for n that produces it for your chosen sequence model.
Yes. It shows the real-valued solution and the nearest integer with the resulting sum error.
Arithmetic series allow negative differences. Geometric series with negative ratios are supported via numeric solving when closed-form is invalid.
Some parameter combinations cannot reach the target sum, e.g., geometric sums with small ratios may have upper bounds below your target.
The bisection method iterates until high precision. For difficult cases, it reports when no finite solution is reachable.