Geometric Sequence Calculator Formula
Understand the math behind the geometric sequence calculator. Each variable explained with a worked example.
Formulas Used
Nth Term
nth_term = a1 * pow(r, n - 1)Sum Finite
sum_finite = r != 1 ? a1 * (1 - pow(r, n)) / (1 - r) : a1 * nSum Infinite
sum_infinite = abs(r) < 1 ? a1 / (1 - r) : 0Variables
| Variable | Description | Default |
|---|---|---|
a1 | First Term (a₁) | 2 |
r | Common Ratio (r) | 3 |
n | Number of Terms (n) | 6 |
How It Works
Geometric Sequence
nth Term
aₙ = a₁ × r^(n-1)
Sum of n Terms
Sₙ = a₁ × (1 - r^n) / (1 - r) (when r ≠ 1)
Infinite Sum (converges only when r
S∞ = a₁ / (1 - r)
A geometric sequence has a constant ratio between consecutive terms.
Worked Example
Geometric sequence: first term = 2, common ratio = 3, 6 terms.
a1 = 2r = 3n = 6
- 01a₆ = 2 × 3⁵ = 2 × 243 = 486
- 02Sum = 2 × (1 - 3⁶) / (1 - 3) = 2 × (1 - 729) / (-2) = 728
- 03Sequence: 2, 6, 18, 54, 162, 486
Frequently Asked Questions
What is a geometric sequence?
A geometric sequence has a constant ratio between consecutive terms. Examples: 2,6,18,54... (r=3) or 100,50,25,12.5... (r=0.5).
When does the infinite sum converge?
The infinite sum converges only when |r| < 1. When |r| ≥ 1, the terms grow without bound and the sum is infinite.
What is the common ratio?
The common ratio r is the factor you multiply by to get from one term to the next: r = a₂/a₁.
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Open Geometric Sequence Calculator