What is Geometric Sequence?

Use this calculator to compute Geometric Sequence values with step-by-step solutions.

How do I use this calculator?

Enter the required values; results and step-by-step derivation update automatically.

Where is Geometric Sequence used?

Sequences, calculus, physics, engineering, and related disciplines.

What formulas does this use?

All standard geometric sequence formulas are applied and shown step-by-step in the results.

Can I get step-by-step solutions?

Yes. The calculator shows a full step-by-step derivation when results are displayed.

How accurate are the results?

Results use standard floating-point arithmetic (~15 significant digits). Suitable for education and professional use.

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ALGEBRASequencesMathematics Calculator

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Geometric Sequences

Each term is the previous multiplied by a constant ratio r. The nth term is aₙ = a₁ × r^(n-1). When |r| < 1, the infinite sum converges to a₁/(1-r).

Concept Fundamentals

aₙ = a₁ × r^(n-1)

nth term

Sₙ = a₁(1-rⁿ)/(1-r)

Finite sum

S∞ = a₁/(1-r) if |r|<1

Infinite sum

|r| < 1

Convergence

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Compound interest is a geometric sequence — balance grows by (1+rate)^n each period. Zeno's paradox uses an infinite geometric series that converges to 1. For |r| ≥ 1 the series diverges; for |r| < 1 it converges to a₁/(1-r).

Key quantities

aₙ = a₁ × r^(n-1)

nth term

Key relation

Sₙ = a₁(1-rⁿ)/(1-r)

Finite sum

Key relation

S∞ = a₁/(1-r) if |r|<1

Infinite sum

Key relation

|r| < 1

Convergence

Key relation

Ready to run the numbers?

Why: Geometric sequences model compound interest, bacterial growth, radioactive decay, and fractal scaling.

How: Identify first term and common ratio; use formulas for nth term and sum. For infinite sum, check |r| < 1.

Compound interest is a geometric sequence — balance grows by (1+rate)^n each period.Zeno's paradox uses an infinite geometric series that converges to 1.

Sources:Khan AcademyWolfram MathWorld

Run the calculator when you are ready.

Exponential PatternsFrom compound interest to radioactive decay

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MATHEMATICS · SEQUENCES

Geometric Sequences — Exponential Patterns

Powers of 2, compound interest, decay, bacterial growth, fractal scaling. Calculate any term or sum instantly.

Arithmetic Sequence →Fibonacci →

📐 Click an Example to Load

Sequence Parameters

First Term (a₁)

Common Ratio (r)

Number of Terms (n)

geometric_sequence.sh

CALCULATED

$ calc_geometric --a1=2 --r=3 --n=10

→ Processing... ✓ Done.

nth Term (aₙ)

39366

Sum (Sₙ)

59048

Infinite Sum

Diverges

Converges?

sequence: a1=2, a2=6, a3=18, a4=54, a5=162, a6=486, a7=1458, a8=4374, a9=13122, a10=39366

Geometric Sequence Calculator

a₁=2, r=3, n=10

39366 (nth term)

∑ Sₙ = 59048

numbervibe.com/calculators/mathematics/sequences/geometric-sequence

📈 Exponential Growth Curve (Line)

📊 Term Comparison (Bar)

🍩 Sum Distribution (Doughnut)

Step-by-Step Solution

We have a geometric sequence with first term a₁ = 2 and common ratio r = 3.
To find the 10th term, we use the formula: a_n = a₁ × r^(n-1)
a_10 = 2 × 3^(10 - 1)
a_10 = 2 × 3^9
a_10 = 2 × 19683
a_10 = 39366
To find the sum of the first 10 terms, we use the formula: S_n = a₁ × (1 - r^n) / (1 - r)
S_10 = 2 × (1 - 3^10) / (1 - 3)
S_10 = 2 × (1 - 59049) / -2
S_10 = 2 × -59048 / -2
S_10 = -118096 / -2
S_10 = 59048

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

💰

Compound interest follows geometric growth: balance = P(1+r)^n.

🦠

Bacterial growth follows geometric sequences when each cell divides into two.

📋 Key Takeaways

• nth term: aₙ = a₁ × r^(n-1) — each term is the previous multiplied by the common ratio
• Constant ratio: r = aₙ₊₁ / aₙ for all n — the hallmark of geometric sequences
• Finite sum: Sₙ = a₁(1-rⁿ)/(1-r) for r≠1; Sₙ = n×a₁ when r=1
• Infinite sum: S∞ = a₁/(1-r) only when |r| < 1 — otherwise the series diverges

💡 Did You Know?

💰Compound interest is a geometric sequence — your balance grows by (1 + rate)^n each periodSource: Finance

🦠Bacterial growth follows geometric sequences when each cell divides into twoSource: Biology

📱Moore's Law (doubling every 18 months) is a geometric progression in techSource: Technology

☢️Radioactive half-life decay is geometric — each half-life halves the remaining amountSource: Physics

🌀Fractal patterns use geometric scaling — each level is r times the previousSource: Geometry

🏃Zeno's paradox (Achilles and the tortoise) uses an infinite geometric series that converges to 1Source: Philosophy

📖 How Geometric Sequences Work

A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a fixed non-zero number called the common ratio (r). Unlike arithmetic sequences (which add a constant), geometric sequences multiply by a constant.

General Form

a₁, a₁r, a₁r², a₁r³, ..., a₁rⁿ⁻¹

The nth term: aₙ = a₁ × r^(n-1)

Convergence

When |r| < 1, the sequence converges to 0 and the infinite sum S∞ = a₁/(1-r) exists. When |r| ≥ 1, the sequence diverges and the infinite sum is undefined.

🎯 Expert Tips

💡 Identifying the Ratio

Divide any term by the previous: r = aₙ₊₁ / aₙ. Works for any consecutive pair.

💡 Convergent vs Divergent

|r| < 1 → converges; |r| > 1 → diverges; r = ±1 → special cases (constant or alternating).

💡 Compound Interest Modeling

Use a₁ = principal, r = 1 + (rate/100). The nth term is your balance after n periods.

💡 Infinite Sum

S∞ = a₁/(1-r) only when |r| < 1. Example: 1 + ½ + ¼ + ... = 2.

⚖️ Why Use This Calculator vs. Other Tools?

Feature	This Calculator	Spreadsheet	Manual
nth term formula	✅	⚠️ Formula needed	❌
Finite sum	✅	✅	❌
Infinite sum (\|r\|<1)	✅	⚠️	❌
Exponential growth chart	✅	✅	❌
Term comparison bar chart	✅	⚠️	❌
Step-by-step solution	✅	❌	❌
7+ preset examples	✅	❌	❌
Copy & share results	✅	❌	❌

❓ Frequently Asked Questions

What is the difference between geometric and arithmetic sequences?

Geometric: multiply by constant r. Arithmetic: add constant d. Example: 2,4,8,16 is geometric (×2); 2,4,6,8 is arithmetic (+2).

Can the common ratio be 1?

Yes. When r=1, every term equals a₁, so Sₙ = n×a₁. The sequence is constant.

How do I find the common ratio from two terms?

Divide: r = aₙ₊₁ / aₙ. Example: a₂=6, a₃=18 → r = 18/6 = 3.

When does the infinite geometric series converge?

Only when |r| < 1. Then S∞ = a₁/(1-r). If |r| ≥ 1, the sum diverges.

Can geometric sequences have fractions or decimals?

Yes. Both a₁ and r can be any real numbers. Example: 0.5, 0.05, 0.005,... with r=0.1.

How do I find the first term from another term and r?

Use a₁ = aₙ / r^(n-1). Example: a₅=96, r=2 → a₁ = 96/2⁴ = 6.

What is a negative common ratio?

Terms alternate in sign. Example: 2, -4, 8, -16 with r=-2.

How is compound interest related to geometric sequences?

Balance after n periods = P(1+r)^n. This is a geometric sequence with a₁=P and common ratio (1+r).

📊 Geometric Sequences by the Numbers

aₙ = a₁rⁿ⁻¹

nth Term Formula

|r| < 1

Convergence Condition

a₁/(1-r)

Infinite Sum

Finance, Bio, Physics

Key Applications

📚 Official & Educational Sources

Khan Academy - Geometric Sequences ↗

Free video lessons on geometric sequences

Updated: 2025

Wolfram MathWorld - Geometric Series ↗

Mathematical reference for geometric series

Updated: 2025

OEIS - Integer Sequences ↗

Online Encyclopedia of Integer Sequences

Updated: 2025

Math is Fun - Geometric Sequences ↗

Beginner-friendly geometric sequence guide

Updated: 2025

⚠️ Disclaimer: This calculator provides mathematical results for educational purposes. For financial, scientific, or engineering applications, verify formulas and results with authoritative sources. Not financial or medical advice.

👈 START HERE

⬅️Jump in and explore the concept!

Geometric Sequences

Geometric Sequences — Exponential Patterns

📐 Click an Example to Load

Sequence Parameters

📈 Exponential Growth Curve (Line)

📊 Term Comparison (Bar)

🍩 Sum Distribution (Doughnut)

Step-by-Step Solution

🧮 Fascinating Math Facts

📋 Key Takeaways

💡 Did You Know?

📖 How Geometric Sequences Work

General Form

Convergence

🎯 Expert Tips

💡 Identifying the Ratio

💡 Convergent vs Divergent

💡 Compound Interest Modeling

💡 Infinite Sum

⚖️ Why Use This Calculator vs. Other Tools?

❓ Frequently Asked Questions

What is the difference between geometric and arithmetic sequences?

Can the common ratio be 1?

How do I find the common ratio from two terms?

When does the infinite geometric series converge?

Can geometric sequences have fractions or decimals?

How do I find the first term from another term and r?

What is a negative common ratio?

How is compound interest related to geometric sequences?

📊 Geometric Sequences by the Numbers

📚 Official & Educational Sources

Related Calculators

Sequence Arithmetic Calculator

Sequence General Calculator

Convolution Calculator

Fibonacci Calculator

Sum Of Series Calculator

Collatz Conjecture Calculator

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Geometric Sequences

Geometric Sequences — Exponential Patterns

📐 Click an Example to Load

Sequence Parameters

📈 Exponential Growth Curve (Line)

📊 Term Comparison (Bar)

🍩 Sum Distribution (Doughnut)

Step-by-Step Solution

🧮 Fascinating Math Facts

📋 Key Takeaways

💡 Did You Know?

📖 How Geometric Sequences Work

General Form

Convergence

🎯 Expert Tips

💡 Identifying the Ratio

💡 Convergent vs Divergent

💡 Compound Interest Modeling

💡 Infinite Sum

⚖️ Why Use This Calculator vs. Other Tools?

❓ Frequently Asked Questions

What is the difference between geometric and arithmetic sequences?

Can the common ratio be 1?

How do I find the common ratio from two terms?

When does the infinite geometric series converge?

Can geometric sequences have fractions or decimals?

How do I find the first term from another term and r?

What is a negative common ratio?

How is compound interest related to geometric sequences?

📊 Geometric Sequences by the Numbers

📚 Official & Educational Sources

Related Mathematics Calculators

Related Calculators

Sequence Arithmetic Calculator

Sequence General Calculator

Convolution Calculator

Fibonacci Calculator

Sum Of Series Calculator

Collatz Conjecture Calculator

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