Geometric Sequence Calculator | nth Term & Sum
Find the nth term, partial sum, and infinite sum of any geometric sequence. Solve for the common ratio or number of terms. Determine convergence with first 12 terms shown.
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What Is the Geometric Sequence Calculator | nth Term & Sum?
A geometric sequence (also called a geometric progression, GP) is an ordered list where each term is obtained by multiplying the previous term by a fixed constant, the common ratio r. Unlike arithmetic sequences that add, geometric sequences multiply. This makes them the mathematical model for exponential growth and decay.
- ▸Find nth Term: Enter a₁, r, and n to compute aₙ = a₁ × r^(n−1) plus the partial and infinite sums.
- ▸Find Sum: Computes Sₙ directly for the given n, and S∞ when |r| < 1.
- ▸Find Ratio: Know a₁, aₙ, and n? Recovers r = (aₙ/a₁)^(1/(n−1)).
- ▸Find n: Given a₁, r, and a target term value, finds what position n it occupies.
- ▸Convergence indicator: Clearly shows whether the series converges or diverges based on |r|.
Formula
A geometric sequence is defined by its first term a₁ and a common ratio r. Every term is r times the previous term.
nth Term Formula
aₙ = a₁ × r^(n−1)
Direct formula, no need to compute all previous terms.
Partial Sum (r ≠ 1)
Sₙ = a₁ × (rⁿ − 1) / (r − 1)
Sum of the first n terms. When r = 1, Sₙ = n × a₁.
Infinite Sum (|r| < 1)
S∞ = a₁ / (1 − r)
Only valid when |r| < 1; otherwise the series diverges.
Recursive Formula
a₁ = given aₙ = aₙ₋₁ × r
Each term is the previous multiplied by r.
How to Use
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Choose a mode
Select Find nth Term, Find Sum, Find Common Ratio, or Find Number of Terms depending on what you want to compute.
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Try a preset
Click Doubling, Half-life, Compound 5%, or another preset to load a real-world example instantly.
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Enter first term (a₁)
The starting value of the sequence. Can be any non-zero real number.
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Enter common ratio (r)
The fixed multiplier. |r| < 1 gives convergence; |r| > 1 gives divergence; negative r gives alternating terms.
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Set n (number of terms)
The term position or length of partial sum you want. Supports up to 500.
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Press Calculate
Results show the nth term, partial sum, infinite sum (if convergent), and first 12 terms as chips.
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Check convergence
The convergence badge tells you whether |r| < 1 (converges) or |r| ≥ 1 (diverges) with the actual |r| value.
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Expand steps
Click Step-by-step working to see every substitution and calculation in numbered detail.
Example Calculation
Example 1 | Half-life decay (a₁ = 1024, r = 0.5, n = 10)
Classic radioactive decay model, amounts halve each period.
Example 2 | Compound interest (a₁ = 1000, r = 1.05, n = 12)
5% annual growth on an initial ’1000, models a savings account balance.
Example 3 | Zeno's paradox (a₁ = 1, r = 0.5)
Demonstrates that an infinite number of diminishing steps can sum to a finite value.
Understanding Geometric Sequence | nth Term & Sum
What Is a Geometric Sequence?
A geometric sequence is an ordered list where every term is produced by multiplying the previous term by the same constant factor r. The sequence 2, 6, 18, 54 has r = 3; the sequence 100, 50, 25 has r = 0.5. All calculations run live in your browser, no server required.
Convergence vs Divergence
The critical question for any infinite geometric series is whether it converges:
- ▸|r| < 1, Terms shrink to zero; infinite sum S∞ = a₁/(1−r) is finite.
- ▸|r| = 1, Terms are constant (r = 1) or alternate ±a₁ (r = −1). Series diverges.
- ▸|r| > 1, Terms grow without bound. Series diverges.
Real-World Applications
- ▸Compound interest: Balance grows by factor (1 + i) each period.
- ▸Radioactive decay: Quantity halves every half-life period.
- ▸Population growth: Geometric model when growth rate is constant.
- ▸Mortgage payments: Present value formulas use infinite geometric series.
- ▸Signal attenuation: Signal power reduces by constant factor each section.
The Infinite Sum Formula
The formula S∞ = a/(1−r) is one of the most-used results in mathematics. It underlies the present-value formula for perpetuities in finance, resolves Zeno's paradox in philosophy, and appears throughout power series in calculus. The derivation: write S = a + ar + ar² + …, then rS = ar + ar² + …, and S − rS = a, giving S(1−r) = a.
Frequently Asked Questions
When does a geometric series converge?
A geometric series converges to a finite sum if and only if the absolute value of the common ratio is less than 1.
- • Condition: |r| < 1
- • Infinite sum: S∞ = a₁ / (1 − r)
- • When |r| ≥ 1, the terms do not approach zero so the series diverges.
What is the common ratio and how do I find it?
The common ratio r is the factor by which each term is multiplied to get the next. Divide any term by the previous term:
- • r = aₙ₊₁ / aₙ
- • If you know a₁, aₙ, and n: r = (aₙ/a₁)^(1/(n−1))
What happens when r is negative?
A negative ratio produces an alternating sequence (positive, negative, positive, …). For example, a₁ = 1, r = −0.5 gives: 1, −0.5, 0.25, −0.125, …
Convergence still depends only on |r|: the series converges when |r| < 1, regardless of sign. The infinite sum formula S∞ = a/(1−r) still applies.
How is a geometric sequence different from an arithmetic sequence?
The fundamental difference is the operation between terms:
- • Arithmetic: each term = previous + d (constant addition)
- • Geometric: each term = previous × r (constant multiplication)
Arithmetic sequences model linear change; geometric sequences model exponential change.
What are real-world examples of geometric sequences?
Geometric sequences appear everywhere exponential growth or decay occurs:
- • Compound interest (r = 1 + interest rate)
- • Radioactive decay (r = e^(−λ), where λ is the decay constant)
- • Bouncing ball height after each bounce (r = coefficient of restitution)
- • Moore's Law, transistor count doubling roughly every 2 years
- • COVID spread, each infected person spreads to r others
How does Zeno's paradox relate to geometric series?
Zeno argued that Achilles can never catch a tortoise because he must first reach where it was, then where it moved to, and so on, an infinite number of steps. The distances form a geometric series with |r| < 1:
- • Steps: d, dr, dr², … with |r| < 1
- • Total: S∞ = d/(1−r), a finite distance
The paradox is resolved because infinite steps can sum to a finite total when the steps shrink fast enough.
Can the first term be zero?
If a₁ = 0, every term is 0 regardless of r, and Sₙ = 0 for all n. This is a trivial (degenerate) geometric sequence. For finding the common ratio from a₁ and aₙ, a₁ = 0 causes a division by zero, so the calculator requires a₁ ≠ 0 in ratio-finding mode.
What is the sum formula when r = 1?
When r = 1, every term equals a₁, and the standard sum formula has a 0 denominator. The sum is simply:
- • Sₙ = n × a₁
- • The series diverges (no finite S∞).
The calculator handles this case automatically.