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.