geometric progression calculator

Calculate the n-th term, sum of first n terms, and infinite sum (when it converges).

First term (a₁)

Common ratio (r)

Number of terms (n)

Enter values and click Calculate All.

What is a geometric progression?

A geometric progression (also called a geometric sequence) is a sequence where each term is obtained by multiplying the previous term by the same constant value, called the common ratio.

For example, in the sequence 2, 6, 18, 54, ... each term is multiplied by 3, so the common ratio is 3. Geometric progressions appear in compound growth, depreciation, population change, and many financial models.

Core formulas

n-th term: a_n = a₁ × r^n-1

Sum of first n terms: S_n = a₁ × (1 - rⁿ) / (1 - r), for r ≠ 1

Special case when r = 1: S_n = n × a₁

Infinite sum: S_∞ = a₁ / (1 - r), only if |r| < 1

How to use this calculator

Enter the first term a₁.
Enter the common ratio r.
Enter the number of terms n (positive integer).
Click Calculate All to get key results instantly.
Click Generate Sequence to list the first n terms.

Why geometric progressions matter

1) Compound interest and investing

If money grows by a fixed percentage each period, the value follows a geometric pattern. That is why even small regular growth can lead to large long-term results.

2) Depreciation

Asset values often decline by a fixed percentage each year. That decrease is also geometric, just with a ratio less than 1.

3) Scientific and data applications

Geometric models are used in epidemiology, signal processing, and algorithm analysis whenever repeated proportional change is involved.

Quick worked example

Suppose a₁ = 500, r = 1.08, and n = 12. The 12th term gives the value after 12 growth steps, and S₁₂ gives the total accumulated value across those first 12 terms. Entering those values in the calculator gives both instantly, plus the infinite sum check.

Common mistakes to avoid

Using n = 0 or a negative number of terms.
Forgetting that infinite sum only exists when |r| < 1.
Mixing up arithmetic vs geometric formulas.
Rounding too aggressively in early steps.

Final note

A geometric progression calculator is a small tool with big utility. Whether you're estimating investment growth, modeling decay, or studying sequences, the right formulas can save time and prevent errors. Use this page anytime you need fast, reliable GP results.