Geometric Series Calculator
Calculate the sum of a geometric series given the first term, common ratio, and number of terms.
Also shows infinite sum when |r| < 1.
A geometric series is a sequence of numbers where each term is found by multiplying the previous term by a fixed constant called the common ratio (r). Unlike arithmetic sequences (which add a constant), geometric sequences multiply — producing exponential growth or decay.
The formula for the sum of a finite geometric series: S_n = a × (1 − r^n) / (1 − r) when r ≠ 1
Where:
- a = first term
- r = common ratio
- n = number of terms
- S_n = sum of first n terms
Sum of an infinite geometric series (only valid when |r| < 1): S_∞ = a / (1 − r)
Worked example — Finite series: Series: 3, 6, 12, 24, 48 (a = 3, r = 2, n = 5) S_5 = 3 × (1 − 2^5) / (1 − 2) = 3 × (1 − 32) / (−1) = 3 × 31 = 93 Verification: 3 + 6 + 12 + 24 + 48 = 93 ✓
Worked example — Infinite series: Series: 1, 0.5, 0.25, 0.125, … (a = 1, r = 0.5) S_∞ = 1 / (1 − 0.5) = 1 / 0.5 = 2 This is why 1 + ½ + ¼ + ⅛ + … forever equals exactly 2.
Real-world applications:
- Compound interest: Your savings grow geometrically (each year multiplied by 1 + interest rate)
- Population growth: Each generation multiplied by the growth factor
- Depreciation: Asset value multiplied by (1 − depreciation rate) each year
- Geometric sequences in nature: Shell spirals, bacterial growth, radioactive decay all follow r values
Key rule: If |r| ≥ 1, the infinite series does not converge — it grows without bound.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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