Mathematics
July 22, 2026 · 10 min read
Geometric Sequences: Exponential Growth and Decay Patterns
Master geometric sequences with formulas, worked examples, and applications in finance, physics, and natural growth patterns.
Try the free calculator
Put these formulas into practice with our instant, step-by-step Geometric Sequence Calculator.
A geometric sequence is an ordered list of numbers where each term is obtained by multiplying the previous term by a constant called the common ratio. Unlike arithmetic sequences (which add a constant), geometric sequences multiply — leading to exponential growth or decay.
The nth Term Formula
aₙ = a₁ × rⁿ⁻¹
Where a₁ is the first term and r is the common ratio.
The Sum Formula
Sₙ = a₁ × (1 - rⁿ) / (1 - r), for r ≠ 1
For infinite geometric series with |r| < 1, the sum converges to S = a₁/(1-r).
Worked Example
Find the 6th term and sum of the sequence 3, 6, 12, 24, ...:
- Identify: a₁ = 3, r = 2
- 6th term: a₆ = 3 × 2⁵ = 3 × 32 = 96
- Sum: S₆ = 3 × (1 - 2⁶)/(1 - 2) = 3 × (-63)/(-1) = 189
Applications
- Compound interest: Money grows geometrically
- Population growth: Bacteria multiply geometrically
- Radioactive decay: Half-life creates geometric decay
- Fractal geometry: Self-similar patterns