Math & Statistics · 等比数列（Geometric Sequence） · Updated 2026-10-25
What Is Geometric Sequence?
Definition
The nth term is an = a1·r^(n−1) with ratio r, and Sn = a1(r^n−1)/(r−1). Its intuitive power lies in exponential growth: doubling 10 times is 2^10 = 1024×, not 20×. The Rule of 72 (years to double ≈ 72 ÷ rate) is an engineering approximation of geometric growth: ln2 ≈ 0.693, nudged to 72 for mental math. The same structure describes decay: depreciation, drug half-lives and radioactive decay are geometric with r<1.
Formula
an = a1·r^(n−1); Sn = a1(r^n−1)/(r−1) for r≠1
Worked Example
At 7% annualized over 30 years: 1.07^30 ≈ 7.6× — 7% a year nearly triples in three decades. And one infects three, three infect nine: 3^10 ≈ 59,000 after ten rounds.
Common Misconception
Using the sum formula blindly near r=1 — at r=1 the denominator is zero and the series degenerates to a constant (Sn = na1). Watch negative ratios too: 1, −2, 4, −8… satisfies the general term but loses monotonicity of partial sums.
Matching Calculators
Related Terms
Keep Exploring
Browse the full glossary, or head back to the homepage for 1,435 free online calculators.
Back to glossary