geometric series calculator - Aaron Graves, PhDude Replica

Calculate geometric sequence terms and sums instantly

Enter the first term, common ratio, and number of terms. This tool returns the n-th term, the finite sum, and whether the infinite sum converges.

First term (a)

The sequence starts with this value.

Common ratio (r)

Each term is multiplied by this ratio to get the next term.

Number of terms (n)

Use a positive whole number.

Enter values and click Calculate to see results.

What is a geometric series?

A geometric series is the sum of terms in a geometric sequence. In a geometric sequence, each term is found by multiplying the previous term by a constant called the common ratio, usually written as r.

For example, if the first term is 3 and the ratio is 2, the sequence is: 3, 6, 12, 24, 48, ... and the corresponding series (sum) is: 3 + 6 + 12 + 24 + 48 + ...

Core formulas
n-th term: a_n = a · r^n-1
Finite sum: S_n = a(1 - rⁿ) / (1 - r), for r ≠ 1
Special case when r = 1: S_n = n · a
Infinite sum (if |r| < 1): S_∞ = a / (1 - r)

How to use this geometric series calculator

Enter the first term of your sequence.
Enter the common ratio.
Enter the number of terms you want to include in the finite sum.
Click Calculate to get all values at once.

The tool also previews the first several terms so you can quickly verify your setup and avoid common input errors.

When does an infinite geometric series converge?

The infinite geometric sum only exists when the ratio is strictly between -1 and 1 in absolute value: |r| < 1.

If |r| < 1, terms shrink toward zero and the infinite sum converges.
If |r| ≥ 1, terms do not shrink enough, and the infinite sum diverges.

This idea appears everywhere in finance, physics, computer science, and signal processing.

Quick real-world examples

Finance: discounted cash flow and perpetuity-style models.
Population models: repeated proportional growth or decay.
Computer graphics: recursive scaling effects.
Learning models: repeated reduction in error over iterations.

Common mistakes to avoid

Using n = 0 (the calculator requires a positive integer for number of terms).
Forgetting that r = 1 uses a special sum formula.
Trying to compute an infinite sum when |r| ≥ 1.
Mixing up sequence values and series sums.

FAQ

Can the ratio be negative?

Yes. A negative ratio causes terms to alternate signs. The formulas still apply.

What if the ratio is 0?

Then all terms after the first are zero. The finite and infinite sums both equal the first term.

Does this support decimals and fractions?

Yes. You can enter decimal values directly (for fractions, use decimal equivalents like 0.25 for 1/4).

Summary

This geometric series calculator gives a fast, reliable way to compute the n-th term, finite sum, and infinite sum behavior from the same inputs. Use it to save time, verify homework, or check assumptions in practical modeling problems.