Geometric Sequence Calculator
Calculate geometric sequences, series sums, and explore exponential patterns
Calculate Geometric Sequence
Sequence Results
General formula: aₙ = a₁ × r^(n-1) = 2 × 3^(n-1)
Explicit formula: aₙ = 2 × 3^(n-1)
Recursive formula: aₙ = aₙ₋₁ × 3, a₁ = 2
Finite Series Sum
Infinite Series Sum
Geometric Sequence Terms
Pattern: Each term = previous term × 3
Formula: aₙ = 2 × 3^(n-1)
Sequence Properties
Common Ratios
Key Formulas
Understanding Geometric Sequences
What is a Geometric Sequence?
A geometric sequence 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).
Key Components
- •First term (a₁): The starting value of the sequence
- •Common ratio (r): The multiplier between consecutive terms
- •Nth term (aₙ): Any term in the sequence at position n
Explicit Formula
aₙ = a₁ × r^(n-1)
where n is the position of the term
Geometric Series
A geometric series is the sum of terms in a geometric sequence. The sum depends on whether the series is finite or infinite, and whether it converges.
Finite Series Sum
S = a₁ × (r^n - 1) / (r - 1)
for r ≠ 1, where n is the number of terms
Infinite Series Sum
S∞ = a₁ / (1 - r)
only when |r| < 1 (convergent series)
Real-World Applications
- 💰Compound interest calculations
- 📈Population growth models
- 🔬Radioactive decay
- 💻Computer memory and binary systems