AkCalculators

Geometric sequences and series — concepts, convergence and applications

Geometric sequences (also known as geometric progressions) are pervasive throughout mathematics and applied fields. They model exponential growth and decay, compound interest, population models under fixed proportional change, and many algorithms' time and space behaviors. A geometric sequence is described by a first term a₁ and a common ratio r. Each subsequent term is obtained by multiplying the previous term by r, generating the sequence: a₁, a₁r, a₁r², a₁r³, ….

Deriving the nth term and partial sums

The nth term follows immediately from repeated multiplication: the k-th term multiplies the first term by r raised to the power (k−1). Therefore

To obtain a closed-form for partial sums, consider Sₙ = a₁ + a₁r + a₁r² + … + a₁r^{n−1}. Multiply Sₙ by r and subtract to eliminate most terms:

Sₙ = a₁ (1 + r + r² + ... + r^{n−1})
 rSₙ = a₁ (r + r² + ... + r^{n})
 Sₙ − rSₙ = a₁ (1 − r^{n})
 Sₙ (1 − r) = a₁ (1 − r^{n})
 Sₙ = a₁ (1 − r^{n}) / (1 − r) (for r ≠ 1)
        

Convergence of the infinite geometric series

The infinite geometric series ∑_{k=0}^{∞} a₁ r^k converges if and only if |r| < 1. In that case r^{n} → 0 as n → ∞ and the sum tends to S_∞ = a₁ / (1 − r). If |r| ≥ 1 the terms do not tend to zero and the infinite sum diverges. This simple criterion makes geometric series one of the few infinite series that can be analyzed fully without advanced tests.

Behavior for different ranges of r

• |r| < 1: Terms decay to 0, partial sums approach a finite limit.
• r = 1: Constant sequence, partial sums grow linearly with n.
• r > 1: Exponential growth — terms and partial sums grow without bound.
• r < −1: Magnitude grows and sign alternates; no convergence.
• −1 < r < 0: Alternating decay — magnitude decays to zero but signs alternate; infinite sum converges if |r| < 1.

Applications in finance and science

In finance, compound interest and annuities use geometric progressions to model periodic proportional changes. For example, an investment growing by a fixed rate per period multiplies by (1 + interest_rate) each period — a geometric sequence. Present value and future value formulas reduce to sums of geometric terms. In physics and engineering, models with fixed proportional change per step or iteration (e.g., attenuation, filtering) naturally produce geometric sequences.

Computational considerations

Computing r^n for large n can overflow floating-point types if r > 1, and underflow if |r| << 1. Numerically stable evaluation of partial sums uses direct formulas rather than iterative summation when possible. For sums of alternating series with cancellation, use higher-precision arithmetic if small residuals matter. This calculator accepts simple fraction input (e.g., 1/2) to avoid rounding surprises and convert to the exact decimal representation prior to evaluation.

Examples and intuition

Example — population model: A population grows by 5% each year. With a₁ = 1000 and r = 1.05, the population after 10 years is 1000 × 1.05^{9}. The total population summed across years uses the geometric partial sum.

Example — infinite discount: A perpetuity paying a fixed amount each period discounted by factor r (0 < r < 1) has present value equal to a₁ / (1 − r) — a geometric infinite sum.

Extensions and related sequences

Geometric sequences connect to exponential functions in continuous models: when time intervals shrink and proportional change per interval is small, geometric growth approaches exponential growth. Also, geometric series are building blocks for solving linear difference equations with constant coefficients.

Practical tips

• Use the formulas tab to verify derivations and the examples tab for quick checks.
• Enter ratios as fractions (1/2) when exact rational representation helps clarity; the tool will parse them.
• For financial uses, prefer decimal rates like 0.05 for 5% and ensure consistent period units.

Geometric sequences are simple yet powerful. This calculator helps you compute terms and sums, check convergence of infinite series, and export results for further analysis. If you need additional features (complex ratios, symbolic derivations, or high-precision arithmetic) tell me which capability to add next.