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?

Arithmetic, 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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Geometric Sequences: Exponential Growth

Each term is the previous times common ratio r: aₙ = a₁r^(n−1). Sum Sₙ = a₁(1−rⁿ)/(1−r) when r≠1. Infinite sum S∞ = a₁/(1−r) when |r|<1.

Concept Fundamentals

aₙ = a₁rⁿ⁻¹

nth term

Sₙ = a₁(1−rⁿ)/(1−r)

Sum

S∞ = a₁/(1−r), |r|<1

Infinite

Compound interest

Use

Did our AI summary help? Let us know.

Compound interest: balance = P(1+r)^n — geometric with a₁=P, ratio=1+r. 1+1/2+1/4+... = 2 — infinite geometric with r=1/2. |r|≥1: infinite sum diverges.

Key quantities

aₙ = a₁rⁿ⁻¹

nth term

Key relation

Sₙ = a₁(1−rⁿ)/(1−r)

Sum

Key relation

S∞ = a₁/(1−r), |r|<1

Infinite

Key relation

Compound interest

Use

Key relation

Ready to run the numbers?

Why: Geometric sequences model exponential growth: compound interest, population, radioactive decay. Ratio r: each term = previous × r. |r|<1 gives convergent infinite sum.

How: nth term: aₙ = a₁×r^(n−1). Sum: Sₙ = a₁(1−rⁿ)/(1−r). Infinite: S∞ = a₁/(1−r) when |r|<1. Common ratio r = a₂/a₁.

Compound interest: balance = P(1+r)^n — geometric with a₁=P, ratio=1+r.1+1/2+1/4+... = 2 — infinite geometric with r=1/2.

Sources:NCTM StandardsKhan Academy Sequences

Run the calculator when you are ready.

Calculate Geometric SequenceFind nth term, sum, or infinite sum

Mode

First Term (a₁)

Common Ratio (r)

Term Number (n)

geometric_sequence.sh

CALCULATED

$ geometric --mode find-term --a1 2 --r 3 --n 5

nth Term

162

Terms shown

Formula

a_5 = 162

Sequence: a1=2, a2=6, a3=18, a4=54, a5=162...

Geometric Sequence Calculator

a_5 = 162

numbervibe.com

Term Distribution

📐 Step-by-Step Breakdown

SETUP

First term (a₁)

Common ratio (r)

METHOD

Term number (n)

RESULT

nth term aₙ

162

aₙ = a₁ × r^(n-1) = 2 × 3^4

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

🧮 Fascinating Math Facts

📐

aₙ = a₁r^(n−1). Sum Sₙ = a₁(1−rⁿ)/(1−r).

💰

Compound interest is geometric: P(1+r)^n.

📋 Key Takeaways

• nth term: aₙ = a₁ × r^(n-1), where a₁ is first term, r is common ratio
• Sum of n terms: Sₙ = a₁(1-rⁿ)/(1-r) when r ≠ 1; Sₙ = n×a₁ when r = 1
• Infinite sum: S∞ = a₁/(1-r) only when |r| < 1
• Common ratio: r = a_n+1/a_n for any consecutive terms
• Each term is a constant multiple of the previous term

💡 Did You Know?

📜Geometric sequences model compound interest, population growth, and radioactive decay.Source: Applied Math

🔢Fibonacci described geometric sequences in Liber Abaci (1202).Source: Mathematical History

📐The infinite sum 1 + 1/2 + 1/4 + ... = 2 (Zeno's paradox).Source: Classical Math

🌍Used in finance for present value and annuity calculations.Source: Finance

📏log_b(x) relates to geometric sequences: b^y = x.Source: Logarithms

✏️Each term is a constant multiple of the previous term.Source: NCTM Standards

📖 How It Works

A geometric sequence: a₁, a₁r, a₁r², a₁r³, ... Each term is a₁ × r^(n-1). The sum of the first n terms uses Sₙ = a₁(1-rⁿ)/(1-r) when r ≠ 1. When |r| < 1, the infinite series converges to a₁/(1-r).

📝 Worked Example: 2, 6, 18, 54... — 5th term and sum

Given: a₁ = 2, r = 3

5th term: a₅ = 2 × 3^4 = 2 × 81 = 162

Sum of 5 terms: S₅ = 2(1-3⁵)/(1-3) = 2(1-243)/(-2) = 2×242/2 = 242

Verification: 2+6+18+54+162 = 242 ✓

🚀 Real-World Applications

💰 Compound Interest

Investment growth: principal × (1+r)^n each period.

📉 Radioactive Decay

Half-life: amount halves each period.

🦠 Population Growth

Exponential growth with constant doubling time.

📺 Viral Spread

Each person infects r others — geometric spread.

🎵 Musical Scales

Frequency ratios in equal temperament.

📊 Annuities

Present value of future payments.

⚠️ Common Mistakes to Avoid

Wrong exponent: aₙ = a₁ × r^(n-1), not r^n. For a₅, exponent is 4.
r = 1: Sum formula Sₙ = a₁(1-rⁿ)/(1-r) is undefined when r=1. Use Sₙ = n×a₁.
Infinite sum when |r| ≥ 1: Series diverges — no finite sum.
Negative r: Terms alternate sign; sum formula still works.
Finding r: r = a₂/a₁, not a₁/a₂.

🎯 Expert Tips

💡 Find r

Divide any term by the previous: r = aₙ₊₁ / aₙ

💡 Find a₁

a₁ = aₙ / r^(n-1) if you know the nth term

💡 Convergence

Infinite sum exists only when |r| < 1

💡 r = 1

All terms equal; Sₙ = n × a₁

📊 Reference Table

Formula	Use When
aₙ = a₁ × r^(n-1)	Finding nth term
Sₙ = a₁(1-rⁿ)/(1-r)	r ≠ 1
Sₙ = n × a₁	r = 1
S∞ = a₁/(1-r)	\|r\| < 1

📐 Quick Reference

a₁

First term

Common ratio

Term count

S∞

Infinite sum

🎓 Practice Problems

1, 2, 4, 8... — 10th term? Answer: 512

Sum of 1 + 1/2 + 1/4 + ... (infinite)? Answer: 2

Sum of first 6 terms of 3, 6, 12...? Answer: 189

100, 50, 25... — 8th term? Answer: 0.78125

❓ FAQ

What is a geometric sequence?

A sequence where each term is the previous term multiplied by a constant (common ratio r).

How do I find the common ratio?

Divide any term by the previous: r = aₙ₊₁ / aₙ.

When does the infinite sum exist?

Only when |r| < 1. Then S∞ = a₁/(1-r).

What if r = 1?

All terms are equal. Sₙ = n × a₁.

Applications?

Compound interest, population growth, radioactive decay, annuities.

Difference from arithmetic sequence?

Geometric multiplies by r; arithmetic adds d.

Can r be negative?

Yes. Terms alternate sign. Sum formula still works.

📌 Summary

Geometric sequences multiply by a constant r. Use aₙ = a₁ × r^(n-1) for the nth term and Sₙ = a₁(1-rⁿ)/(1-r) for the sum when r ≠ 1. When |r| < 1, the infinite sum converges to a₁/(1-r).

✅ Verification Tip

Check that aₙ₊₁ / aₙ = r for consecutive terms. For sum, add the first few terms manually and compare.

🔗 Next Steps

Explore the Arithmetic Sequence Calculator for additive growth, or the Summation Calculator for sigma notation. The Logarithm Calculator relates to geometric sequences.

⚠️ Disclaimer: This calculator is for educational purposes. Verify critical calculations independently.

👈 START HERE

⬅️Jump in and explore the concept!

Geometric Sequences: Exponential Growth

Term Values

Term Distribution

📐 Step-by-Step Breakdown

🧮 Fascinating Math Facts

📋 Key Takeaways

💡 Did You Know?

📖 How It Works

📝 Worked Example: 2, 6, 18, 54... — 5th term and sum

🚀 Real-World Applications

💰 Compound Interest

📉 Radioactive Decay

🦠 Population Growth

📺 Viral Spread

🎵 Musical Scales

📊 Annuities

⚠️ Common Mistakes to Avoid

🎯 Expert Tips

💡 Find r

💡 Find a₁

💡 Convergence

💡 r = 1

📊 Reference Table

📐 Quick Reference

🎓 Practice Problems

❓ FAQ

What is a geometric sequence?

How do I find the common ratio?

When does the infinite sum exist?

What if r = 1?

Applications?

Difference from arithmetic sequence?

Can r be negative?

📌 Summary

✅ Verification Tip

🔗 Next Steps

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