Geometric Series Calculator

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). Arithmetic sequences add a constant; geometric ones multiply by it, which is what turns them into 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: the infinite sum exists only when |r| < 1. What happens otherwise depends on r:

• |r| > 1: the terms grow, so the partial sums run off to infinity.
• r = 1: every term is the same, so the sum after n terms is just a × n and never settles.
• r = −1: the terms flip sign forever. The partial sums bounce between a and 0. Nothing grows without bound here, but nothing converges either, which is why the rule is stated as “does not converge” rather than “grows without bound”.

Reading the finite sum as a fraction of the limit. When |r| < 1, the part still missing after n terms is exactly a·rⁿ / (1 − r). That is the same shape as the sum itself, just scaled down by rⁿ. With r = 0.5 you close half the remaining gap with every term, so after 10 terms you are within 0.1% of the limit and after 20 you are within a millionth. With r = 0.99 you need roughly 700 terms to get halfway there. The calculator prints how far along you are, because “the series converges” and “the series has converged by term 10” are very different statements.