Geometric Sequence Calculator
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Understanding Geometric Sequences
Basic Concepts
A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
nth Term: aₙ = a₁rⁿ⁻¹
Sum of n Terms (r ≠ 1): Sₙ = a₁(1-rⁿ)/(1-r)
Sum of n Terms (r = 1): Sₙ = na₁
Sum of Infinite Terms (|r| < 1): S∞ = a₁/(1-r)
Common Ratio: r = aₙ₊₁/aₙ
Properties and Characteristics
Key Properties
- Constant ratio between terms
- Exponential growth/decay pattern
- Geometric mean property
- Terms form exponential function
- Product of equidistant terms
- Convergence conditions
Special Cases
- Unit ratio (r = 1)
- Negative ratio sequences
- Alternating sequences
- Convergent series (|r| < 1)
- Divergent series (|r| > 1)
- Oscillating sequences
Advanced Topics
Applications
- Compound Interest
- Population Growth
- Radioactive Decay
- Sound Wave Attenuation
- Fractals and Self-Similarity
- Musical Harmonics
- Market Growth Models
- Probability Distributions
Mathematical Analysis
Geometric Mean: GM = ⁿ√(a₁×a₂×...×aₙ)
Term Position: n = log_r(aₙ/a₁) + 1
Product Formula: aₖ×aₙ₋ₖ₊₁ = a₁×aₙ
Partial Products: Pₙ = a₁ⁿ×r^(n(n-1)/2)
Related Concepts
- Power Series
- Taylor Series
- Complex Geometric Series
- Infinite Products
- Recursive Sequences
- Continued Fractions
Convergence Theory
Ratio Test: lim|aₙ₊₁/aₙ| = |r| < 1
Root Test: lim|aₙ|^(1/n) = |r| < 1
Absolute Convergence: Σ|aₙ| < ∞