Understanding convergence and divergence
Whether a geometric series converges to a finite sum as more terms are added depends entirely on the magnitude of the common ratio, as summarized below.
| Common ratio |r| | Behavior of terms | Sum to infinity |
|---|---|---|
| |r| < 1 (e.g. r = 0.5) | Terms shrink toward 0 | Converges: S∞ = a₁ ÷ (1 − r) |
| |r| = 1 | Terms stay constant in magnitude (or alternate sign if r = −1) | Diverges (no finite sum, except the trivial a₁ = 0 case) |
| |r| > 1 (e.g. r = 3) | Terms grow without bound in magnitude | Diverges (no finite sum) |
- The sequence preview shows up to the first 8 terms; for n greater than 8, an ellipsis (…) indicates further terms exist, though the nth term and finite sum are computed exactly for the full n entered.
- The infinite-sum result is shown only when |r| < 1; for all other ratios, the series has no finite sum and the calculator indicates this rather than showing a misleading number.
- A common ratio of exactly 0 is not a standard geometric sequence (every term after the first would be 0), and negative ratios produce a sequence that alternates in sign each term.
What is a geometric sequence?
A geometric sequence (or geometric progression) is an ordered list of numbers in which each term after the first is obtained by multiplying the previous term by a constant, called the common ratio (r). For example, 2, 6, 18, 54, 162 is a geometric sequence with first term 2 and common ratio 3, because each term is 3 times the one before it.
When |r| > 1, the terms grow without bound in magnitude (exponential growth); when 0 < |r| < 1, the terms shrink toward zero (exponential decay); when r = 1, every term equals the first term; and when r is negative, the sign of the terms alternates. Geometric sequences model compound interest, population doubling, radioactive decay, and any process where a quantity is repeatedly scaled by the same factor.
A geometric series is the sum of the terms of a geometric sequence. Unlike an arithmetic series, a geometric series can have a finite sum even when extended to infinitely many terms, provided the common ratio's absolute value is strictly less than 1 — a property with no arithmetic-sequence analogue.
How to use this geometric sequence calculator
- Enter the first term of the sequence (a₁) — the starting value.
- Enter the common ratio (r) — the fixed factor each term is multiplied by to get the next term. Use a value between −1 and 1 (excluding 0) to see a convergent infinite sum.
- Enter the number of terms (n) you want to find the nth term and partial sum for.
- Read the nth term, the sum of the first n terms, the sum to infinity (shown only when |r| < 1), and a preview of up to the first 8 terms.
The geometric sequence formulas
The nth term is found by multiplying the first term by the common ratio raised to the power (n − 1): aₙ = a₁ × r^(n−1). Worked example with a₁ = 2, r = 3, n = 5: a₅ = 2 × 3⁴ = 2 × 81 = 162.
The sum of the first n terms uses the closed-form formula Sₙ = a₁ × (rⁿ − 1) ÷ (r − 1) for r ≠ 1, derived by subtracting a scaled copy of the series from itself to cancel all but the first and last terms. When r = 1, every term equals a₁, so the sum is simply Sₙ = a₁ × n. Worked example: S₅ = 2 × (3⁵ − 1) ÷ (3 − 1) = 2 × (243 − 1) ÷ 2 = 2 × 242 ÷ 2 = 242.
When |r| < 1, the term rⁿ shrinks toward 0 as n grows without bound, so the partial-sum formula converges to a finite limit: S∞ = a₁ ÷ (1 − r). For example, with a₁ = 8 and r = 0.5, S∞ = 8 ÷ (1 − 0.5) = 16 — the infinite sum of 8, 4, 2, 1, 0.5, … approaches but never exceeds 16. When |r| ≥ 1, the infinite sum diverges (has no finite value) and this calculator does not report one.
Common mistakes
- Using n instead of (n − 1) as the exponent — the first term (n = 1) is a₁ × r⁰ = a₁, not a₁ × r¹, so the exponent is always (n − 1).
- Applying the infinite-sum formula when |r| ≥ 1 — the series diverges in that case and has no finite sum; the formula a₁ ÷ (1 − r) only applies when |r| < 1.
- Confusing the common ratio with the common difference — geometric sequences multiply by r each step; arithmetic sequences add d each step.
- Forgetting that a negative common ratio produces an alternating sequence (e.g. 2, −6, 18, −54 for a₁ = 2, r = −3), which some learners mistake for an error rather than expected behavior.
자주 묻는 질문
How do you find the nth term of a geometric sequence?
Use the formula aₙ = a₁ × r^(n−1), where a₁ is the first term, r is the common ratio, and n is the term number. For a₁ = 2, r = 3, n = 5: a₅ = 2 × 3⁴ = 2 × 81 = 162.
How do you find the sum of a geometric series?
For a finite number of terms, use Sₙ = a₁ × (rⁿ − 1) ÷ (r − 1) when r ≠ 1. For a₁ = 2, r = 3, n = 5: S₅ = 2 × (243 − 1) ÷ 2 = 242. For an infinite geometric series with |r| < 1, use S∞ = a₁ ÷ (1 − r) instead.
When does an infinite geometric series have a sum?
An infinite geometric series converges to a finite sum only when the absolute value of the common ratio is strictly less than 1 (|r| < 1). In that case, S∞ = a₁ ÷ (1 − r). If |r| ≥ 1, the terms do not shrink toward zero, so the sum grows without bound and the series diverges.
What is the difference between a geometric and an arithmetic sequence?
A geometric sequence multiplies by a constant ratio (r) to get each next term, producing exponential growth or decay. An arithmetic sequence adds a constant difference (d) to get each next term, producing linear growth or decay. A sequence cannot generally be both unless the common ratio is 1 (constant sequence).
Can the common ratio be negative?
Yes. A negative common ratio produces an alternating sequence, where the sign flips each term. For example, a₁ = 2 and r = −3 gives 2, −6, 18, −54, 162, alternating between positive and negative values while still following aₙ = a₁ × r^(n−1).
참고 자료
- NIST Digital Library of Mathematical Functions (DLMF), §1.2 Elementary Algebra: Geometric Progression. dlmf.nist.gov.
- Rosen KH. Discrete Mathematics and Its Applications. 8th ed. McGraw-Hill, 2018. (Sequences and series.)
- Stewart J. Calculus: Early Transcendentals. 8th ed. Cengage Learning, 2015. (Geometric series convergence.)