What happens when r is between 0 and 1?

The terms decrease toward zero. For example, a₁=100, r=0.5 gives 100, 50, 25, 12.5, … — this models radioactive decay or depreciating assets.

What is an infinite geometric series?

When |r| < 1, the sum of infinitely many terms converges: S∞ = a₁ / (1−r). This is not computed here, but for a₁=1, r=0.5: S∞ = 2.

Yes. A negative r creates alternating sequences. For example, a₁=1, r=−2 gives 1, −2, 4, −8, 16, …

How does compound interest relate to geometric sequences?

Compound interest is a geometric sequence where a₁ = principal, r = (1 + rate), and each term is the balance after one compounding period.

What is the common ratio if I know two consecutive terms?

r = aₙ₊₁ / aₙ. Divide any term by the previous term to find the common ratio.

Math

↵ to calculate

Geometric Sequence Calculator - nth Term Sum Online

Calculate the nth term, sum, and first 10 terms of a geometric sequence given the first term and common ratio.

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How to use this calculator

👉 Fill in the boxes below and your answer appears instantly — no maths needed, we do it all for you! 🎉

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In plain English — what does this do?

🔢 This tool does the maths for you! Just type in your numbers, and it gives you the answer right away. No need to count on your fingers or use a pen and paper.

📌 Direct Answer & Summary

A geometric sequence (geometric progression) is a sequence where each term is obtained by multiplying the previous term by a fixed number called the common ratio. Geometric sequences model exponential growth and decay, compound interest, population growth, and many natural phenomena.

First Term (a₁)

Common Ratio (r)

Term Number (n)

What is Geometric Sequence Calculator - nth Term Sum Online?

How to use it

1️⃣ Enter the first term (a₁) of the sequence.
2️⃣ Enter the common ratio (r) — greater than 1 for growth, between 0 and 1 for decay, negative for alternating.
3️⃣ Enter n — the term number you want to calculate.
4️⃣ The nth term, sum of first n terms, and the first 10 terms appear instantly.

Formula

nth term: aₙ = a₁ × rⁿ⁻¹. Sum of n terms: Sₙ = a₁ × (1 − rⁿ) / (1 − r) for r ≠ 1; Sₙ = n × a₁ for r = 1.

💡 See it in action — a real example

For a₁ = 3, r = 2, n = 6: a₆ = 3 × 2⁵ = 96. S₆ = 3 × (1−64)/(1−2) = 3 × 63 = 189. First 10 terms: 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536.

❓ Common questions

What happens when r is between 0 and 1?: The terms decrease toward zero. For example, a₁=100, r=0.5 gives 100, 50, 25, 12.5, … — this models radioactive decay or depreciating assets.
What is an infinite geometric series?: When |r| < 1, the sum of infinitely many terms converges: S∞ = a₁ / (1−r). This is not computed here, but for a₁=1, r=0.5: S∞ = 2.
Can r be negative?: Yes. A negative r creates alternating sequences. For example, a₁=1, r=−2 gives 1, −2, 4, −8, 16, …
How does compound interest relate to geometric sequences?: Compound interest is a geometric sequence where a₁ = principal, r = (1 + rate), and each term is the balance after one compounding period.
What is the common ratio if I know two consecutive terms?: r = aₙ₊₁ / aₙ. Divide any term by the previous term to find the common ratio.

⚡ Related Math Tools & Calculators

Arithmetic Sequence Sequence Calculator Compound Interest