Geometric Sequence Calculator
Geometric Sequence Calculator
Calculate nth term, sum of terms, common ratio, and generate geometric sequences with step-by-step solutions.
Geometric Sequence Result
162
Formula Applied:
Step-by-Step Calculation:
Sequence Analysis:
Sequence Visualization:
Geometric sequences multiply by constant ratio to get next term.
What is a Geometric Sequence?
A geometric sequence (or 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. Geometric sequences are fundamental in mathematics, finance, computer science, and many real-world applications.
Geometric Sequence Formulas
Nth Term Formula
Find any term
Most common formula
Sum of First n Terms
Finite sum (r≠1)
Geometric series sum
Infinite Sum
|r| < 1 only
Convergent series
Common Ratio
Find ratio
From consecutive terms
Geometric Sequence Rules and Properties
1. Basic Properties
Key characteristics of geometric sequences:
• Constant ratio between terms
• Exponential growth/decay
• Can be finite or infinite
• Ratio can be positive or negative
2. Growth Patterns
Behavior based on common ratio:
• r > 1: Exponential growth
• 0 < r < 1: Exponential decay
• -1 < r < 0: Alternating decay
• r < -1: Alternating growth
• r = 1: Constant sequence
3. Special Cases
Important geometric sequence scenarios:
• r = 0: Sequence becomes 0 after first term
• a₁ = 0: All terms are 0
• |r| < 1: Convergent infinite series
• r = -1: Alternating ±a₁
Real-World Applications
Finance & Economics
- Compound interest: Investment growth over time with fixed interest rate
- Depreciation: Asset value decreasing by constant percentage each year
- Population growth: Populations growing at constant percentage rate
- Inflation calculations: Prices increasing by fixed percentage annually
Science & Engineering
- Radioactive decay: Half-life calculations in nuclear physics
- Bacterial growth: Microorganism population doubling
- Signal processing: Digital filters and signal amplification
- Fractal geometry: Self-similar patterns at different scales
Computer Science & Technology
- Binary search: Search space halves each iteration
- Recursive algorithms: Problems reduced by constant factor
- Data compression: Geometric sequences in compression algorithms
- Network routing: Exponential backoff algorithms
Everyday Life
- Chain emails: Messages forwarded to multiple recipients
- Social media sharing: Viral content spread
- Multilevel marketing: Commission structures
- Sports tournaments: Elimination rounds
Common Geometric Sequence Examples
| First Term (a₁) | Common Ratio (r) | First 5 Terms | Application |
|---|---|---|---|
| 1 | 2 | 1, 2, 4, 8, 16 | Binary doubling, cell division |
| 1000 | 0.9 | 1000, 900, 810, 729, 656.1 | 10% depreciation annually |
| 3 | -2 | 3, -6, 12, -24, 48 | Alternating pattern, signal processing |
| 0.5 | 3 | 0.5, 1.5, 4.5, 13.5, 40.5 | Exponential growth models |
Geometric Sequence Formulas Table
| What to Find | Formula | Conditions | Example |
|---|---|---|---|
| Nth term | aₙ = a₁ × r^(n-1) | Any n ≥ 1 | a₁=2, r=3, n=4 → 54 |
| Sum of first n terms | Sₙ = a₁(1-rⁿ)/(1-r) | r ≠ 1 | Sum of 2,6,18,54 = 80 |
| Infinite sum | S = a₁/(1-r) | |r| < 1 | 1 + 1/2 + 1/4 + ... = 2 |
| Common ratio | r = aₙ/aₙ₋₁ | aₙ₋₁ ≠ 0 | 6/2 = 3, 18/6 = 3 |
Step-by-Step Calculation Examples
Example 1: Find 6th term of sequence 2, 6, 18, ...
- Identify first term: a₁ = 2
- Calculate common ratio: r = 6 ÷ 2 = 3
- Use nth term formula: aₙ = a₁ × r^(n-1)
- Plug in values: a₆ = 2 × 3^(6-1)
- Calculate: a₆ = 2 × 3⁵ = 2 × 243 = 486
- Result: The 6th term is 486
Example 2: Sum of first 5 terms with a₁=5, r=2
- Identify values: a₁ = 5, r = 2, n = 5
- Use sum formula: Sₙ = a₁(1 - rⁿ)/(1 - r)
- Plug in values: S₅ = 5(1 - 2⁵)/(1 - 2)
- Calculate powers: 2⁵ = 32
- Compute: S₅ = 5(1 - 32)/(1 - 2) = 5(-31)/(-1) = 155
- Result: Sum of first 5 terms is 155
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Frequently Asked Questions (FAQs)
Q: What's the difference between arithmetic and geometric sequences?
A: Arithmetic sequences add a constant difference (d) to get next term, while geometric sequences multiply by a constant ratio (r). Arithmetic: linear growth. Geometric: exponential growth.
Q: Can the common ratio be zero or negative?
A: Yes! r = 0 gives sequence: a₁, 0, 0, 0,... Negative r gives alternating positive/negative terms. Example: a₁=2, r=-2 gives: 2, -4, 8, -16, 32,...
Q: When does an infinite geometric series converge?
A: An infinite geometric series converges only when |r| < 1. The sum is S = a₁/(1-r). For |r| ≥ 1, the series diverges (sum goes to infinity).
Q: How do I find the common ratio from sequence terms?
A: Divide any term by its previous term: r = aₙ/aₙ₋₁. For example, if sequence is 3, 12, 48, then r = 12/3 = 4 or r = 48/12 = 4.
Master geometric sequence calculations with Toolivaa's free Geometric Sequence Calculator, and explore more mathematical tools in our Math Calculators collection.