geometric sequence calculator

Enter the first term, common ratio, and number of terms to calculate the sequence, the nth term, and the finite sum.

First term (a₁)

Common ratio (r)

Number of terms (n)

Tip: You can use negative or decimal ratios (e.g., -2, 0.5).

What Is a Geometric Sequence?

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by the same value, called the common ratio. If the first term is a₁ and ratio is r, the sequence looks like this:

a₁, a₁r, a₁r², a₁r³, ...

Examples:

2, 6, 18, 54, ... has ratio 3
100, 50, 25, 12.5, ... has ratio 0.5
5, -10, 20, -40, ... has ratio -2

Core Formulas

1) n-th Term Formula

a_n = a₁ · r^{(n - 1)}

Use this to find any specific term in the sequence without listing all previous terms.

2) Finite Sum Formula

S_n = a₁(1 - rⁿ) / (1 - r), r ≠ 1

If r = 1, every term is the same, so:

S_n = n · a₁

3) Infinite Geometric Sum (Convergent Case)

If |r| < 1, the sequence approaches a limit and infinite sum exists:

S_∞ = a₁ / (1 - r)

If |r| ≥ 1, the infinite sum does not converge.

How to Use This Calculator

Enter your first term a₁.
Enter your common ratio r.
Enter the number of terms n (must be a positive integer).
Click Calculate to get:
- the n-th term,
- the sum of the first n terms,
- the infinite sum when applicable,
- and a preview of sequence terms.

Worked Examples

Example A: Growth Sequence

Suppose a₁ = 3, r = 2, and n = 5.

Sequence: 3, 6, 12, 24, 48
5th term: 48
Sum: 3 + 6 + 12 + 24 + 48 = 93

Example B: Decay Sequence

Let a₁ = 80, r = 0.5, n = 6.

Sequence: 80, 40, 20, 10, 5, 2.5
6th term: 2.5
Finite sum: 157.5
Infinite sum: 160 (because |0.5| < 1)

Why Geometric Sequences Matter

Geometric progression appears in many real-world problems:

Finance: compound interest, loan growth, investment projections
Science: population dynamics, radioactive decay
Computer Science: algorithmic scaling and binary tree levels
Everyday Planning: repeated growth/decline patterns in savings or expenses

Common Mistakes to Avoid

Confusing arithmetic and geometric formulas.
Using the finite sum formula with r = 1 without simplification.
Forgetting that the infinite sum only works when |r| < 1.
Entering a non-integer value for n.

Final Thoughts

A good geometric sequence calculator saves time and reduces algebra mistakes. Use the tool above to quickly analyze growth, decay, and totals. If you're learning sequence formulas, try a few manual checks with the displayed terms so the formulas become intuitive.