Geometric Sequence Calculator
Calculate n-th term, sum, and generate sequence terms for geometric progressions
Starting value of the sequence
Multiplier between consecutive terms
Which term to find
How many terms to include
Quick Examples
Term # (a) Sum of Terms (S) Geometric Sequence
Step-by-Step Calculation
n-th Term
a
Sum of Terms
S
Common Ratio
r
Infinite Sum
Common Geometric Sequences
| Name | First Term (a₁) | Ratio (r) | Sequence |
|---|---|---|---|
| Powers of 2 | 1 | 2 | 1, 2, 4, 8, 16... |
| Powers of 10 | 1 | 10 | 1, 10, 100, 1000... |
| Half-life Decay | 100 | 0.5 | 100, 50, 25, 12.5... |
| Powers of 3 | 1 | 3 | 1, 3, 9, 27, 81... |
| Alternating | 1 | -2 | 1, -2, 4, -8, 16... |
| 10% Growth | 100 | 1.1 | 100, 110, 121, 133.1... |
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About Geometric Sequence Calculator
What is a Geometric Sequence?
A geometric sequence (also called a geometric progression) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r).
Example: 2, 6, 18, 54, 162... is a geometric sequence with first term a₁ = 2 and common ratio r = 3.
How to Use This Calculator
- Select a calculation mode - Choose whether to find the n-th term, calculate the sum, or generate a sequence
- Enter the first term (a₁) - The starting value of your sequence
- Enter the common ratio (r) - The multiplier between consecutive terms
- Enter n - The term position or number of terms
- View results - See the calculated value with step-by-step explanation
Key Formulas
n-th Term Formula
aₙ = a₁ × r^(n-1)
Where:
- aₙ = the n-th term
- a₁ = the first term
- n = the term position
- r = the common ratio
Sum of n Terms Formula (Finite Series)
Sₙ = a₁ × (1 - r^n) / (1 - r) when r ≠ 1
Sₙ = n × a₁ when r = 1
Where:
- Sₙ = sum of the first n terms
- n = number of terms
- a₁ = first term
- r = common ratio
Infinite Sum Formula
S∞ = a₁ / (1 - r) when |r| < 1
Common Applications
| Application | Example |
|---|---|
| Compound interest | Money growing at fixed rate |
| Population growth | Bacteria doubling |
| Radioactive decay | Half-life calculations |
| Depreciation | Asset value decreasing by percentage |
| Musical frequencies | Octave intervals |
Examples
Finding the 6th Term
For sequence starting at 3 with r = 2: a₆ = 3 × 2^(6-1) = 3 × 32 = 96
Finding the Sum of First 5 Terms
For sequence 2, 6, 18, 54, 162 (r = 3): S₅ = 2 × (1 - 3^5) / (1 - 3) = 2 × (1 - 243) / (-2) = 242
Frequently Asked Questions
What's the difference between arithmetic and geometric sequences?
An arithmetic sequence has a constant difference between terms (add the same value). A geometric sequence has a constant ratio between terms (multiply by the same value).
Can the common ratio be negative?
Yes! A negative ratio causes the sequence to alternate between positive and negative values. For example: 2, -6, 18, -54... has r = -3.
Can the common ratio be a fraction?
Yes! A ratio between -1 and 1 (excluding 0) creates a sequence that converges toward zero. For example: 100, 50, 25, 12.5... has r = 0.5.
When does an infinite sum exist?
An infinite geometric series has a finite sum only when |r| < 1. This is because the terms get progressively smaller and approach zero.
Tip: Geometric sequences appear everywhere in nature and finance—from spiral shells to compound interest calculations!