Geometric Sequence Calculator

How to Use This Calculator

1. Enter first term a₁, common ratio r, and n.
2. The nth term and partial sum Sₙ are computed instantly.
3. Use Sum to Infinity tab to find S∞ when |r|<1.
4. Use Professional for convergence analysis and application notes.

Formula

Example

Frequently Asked Questions

• What is a geometric sequence? ▼
  A geometric sequence is an ordered list of numbers where each term is obtained by multiplying the previous term by a fixed constant called the common ratio r. The general form is: a₁, a₁r, a₁r², a₁r³, …, and the nth term formula is aₙ = a₁ × r^(n−1). Example: 2, 6, 18, 54, 162, … has a₁ = 2 and r = 3, so a₅ = 2 × 3⁴ = 162. To identify a geometric sequence, check that each term divided by the previous is constant. A common pitfall is confusing geometric (multiplication) with arithmetic (addition). Also note: if r is negative, terms alternate in sign — for example 4, −8, 16, −32, … with r = −2.
• What is the sum formula for a finite geometric series? ▼
  The sum of the first n terms of a geometric series is Sₙ = a₁ × (1 − rⁿ) / (1 − r), valid when r ≠ 1. When r = 1, all terms are equal and Sₙ = n × a₁. Example: a₁ = 2, r = 3, n = 6. S₆ = 2 × (1 − 3⁶) / (1 − 3) = 2 × (1 − 729) / (−2) = 2 × (−728) / (−2) = 728. Cross-check: 2 + 6 + 18 + 54 + 162 + 486 = 728. A common mistake is applying the formula with r = 1 — division by zero occurs, so use Sₙ = n·a₁ in that case. Also be careful with negative ratios: the formula still works, but intermediate values can be confusing.
• When does an infinite geometric series converge? ▼
  An infinite geometric series S∞ = a₁ + a₁r + a₁r² + … converges if and only if |r| < 1 (the absolute value of the ratio is strictly less than 1). In that case the sum is S∞ = a₁ / (1 − r). Example: a₁ = 1, r = 0.5. S∞ = 1 / (1 − 0.5) = 2. Intuition: 1 + 0.5 + 0.25 + 0.125 + … → 2. If |r| ≥ 1, the terms do not shrink to zero and the series diverges. Example: a₁ = 1, r = 2 gives 1 + 2 + 4 + 8 + … → ∞. A common mistake is trying to use the S∞ formula when |r| ≥ 1, which yields a meaningless negative or very large number. Always check |r| < 1 first.
• How is a geometric sequence different from compound interest? ▼
  Compound interest is structurally a geometric sequence. If you invest $P at annual rate i, the balance after n years is P × (1 + i)ⁿ⁻¹ — exactly the geometric formula with a₁ = P and r = 1 + i. Example: $1,000 at 5% annual rate grows as $1,000, $1,050, $1,102.50, $1,157.63, … — a geometric sequence with r = 1.05. The difference is context: finance calculators add payment streams (PMT), present value (PV), and future value (FV), while a geometric sequence calculator handles the pure mathematical pattern. A common pitfall is applying geometric formulas without adjusting for payment frequency — monthly compounding uses r = 1 + i/12 per period, not r = 1 + i.
• What are real-world examples of geometric sequences? ▼
  Geometric sequences model any situation where a quantity is repeatedly multiplied by a fixed factor. Radioactive decay: a substance with half-life 10 years has amounts 1000, 500, 250, 125, … (r = 0.5 every 10 years). Bacterial growth: doubling every hour gives 1, 2, 4, 8, 16, … (r = 2). Compound interest: balance grows by factor (1 + rate) each period. Bouncing ball: if each bounce reaches 70% of the previous height, heights form a geometric sequence with r = 0.7. Drug dosage accumulation: each dose adds a₁ to the remaining fraction of the previous dose. A common pitfall is assuming real growth is perfectly geometric — in practice, populations face resource limits and growth rates change, making logistic models more accurate for long-term projections.

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