Geometric Sequence Calculator
How It Works
This geometric sequence calculator uses established formulas to provide accurate results.
The basic rule:
- aₙ = a₁ · r^(n-1)
- Sₙ = a₁(1 - rⁿ) / (1 - r) (r ≠ 1)
- S∞ = a₁ / (1 - r) if |r| < 1
Results are estimates. Consult a professional for critical decisions.
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Explore Allday Everyday Math →Frequently Asked Questions
What is a geometric sequence?
A geometric sequence is a list of numbers where each term is multiplied by the same value, called the common ratio r. For example: 2, 6, 18, 54 has r = 3.
How do I find the common ratio?
Divide any term by the previous term: r = a₂/a₁. In a geometric sequence, this ratio is constant. It can be positive, negative, or a fraction.
What is an infinite geometric series?
If |r| < 1, the sum of infinitely many terms converges to S∞ = a₁/(1-r). For example, 1 + 1/2 + 1/4 + 1/8 + ... = 1/(1-0.5) = 2. If |r| ≥ 1, the infinite series diverges.
How are geometric sequences used in real life?
Compound interest, population growth, radioactive decay, and bouncing ball heights all follow geometric sequences. Any situation with constant percentage growth or decay is geometric.