Geometric Sequence and Series Calculator

Calculate geometric sequence terms, finite sums, and infinite sums when the common ratio converges.

SequencesSeriesGeometric

Calculate

First term a₁

Common ratio r

Number of terms n

Please enter valid values.

What this calculator teaches

A geometric sequence changes by multiplying by the same ratio each time. It can grow, decay, alternate signs, or converge toward a finite infinite sum.

This is important in compound growth, radioactive decay, finance, computer algorithms, and infinite series.

Formula & Symbols

Concept	Formula	Meaning
nth term	`a_n = a_1 r^{n-1}`	Finds the term in position n.
finite sum	`S_n = a_1(1-r^n)/(1-r)`	Adds the first n terms when r ≠ 1.
infinite sum	`S_∞ = a_1/(1-r), \|r\|<1`	Only valid when the absolute value of r is less than 1.

Step-by-Step Examples

Example 1

Convergent series

Let a₁=3 and r=0.5.
The terms are 3, 1.5, 0.75, ...
Because |r| < 1, the infinite sum exists.

✓ S∞ = 3/(1 - 0.5) = 6.

Example 2

Growing sequence

Let a₁=2 and r=3.
The 5th term is 2·3⁴.
2·81 = 162.

✓ The 5th term is 162.

Where students use this

🎓

High school and college

Use the tool to check homework and understand each step.

📊

Data and modeling

Connect formulas to tables, graphs, and real values.

🧪

Science and engineering

Use math results inside physics, chemistry, and engineering problems.

💻

Computer science

Apply the same logic to algorithms, systems, and numerical work.

Common Mistakes to Avoid

⚠️

Using arithmetic formulas

Geometric sequences multiply by r; arithmetic sequences add d.

⚠️

Applying infinite sum when |r| ≥ 1

The infinite sum formula only works when the terms shrink toward 0.

⚠️

Forgetting r can be negative

A negative ratio alternates signs between terms.

Frequently Asked Questions

How do I know if a sequence is geometric? ▾

Divide consecutive terms. If the ratio is constant, the sequence is geometric.

What is common ratio? ▾

It is the multiplier used to get from one term to the next.

When does an infinite geometric series converge? ▾

It converges only when |r| < 1.

What happens when r = 1? ▾

Every term is the same, so Sₙ = n·a₁.

Related Math Calculators

Formula Explorer connections

Interpretation: This formula describes an algebraic relationship among variables, functions, equations, roots or sequences. Assumption: Respect the expression’s domain and excluded values. Check roots in the original equation because transformations can introduce extraneous solutions.

Heat Equation Approximation Calculator →Hyperbola Equation Calculator →Inequality Calculator →Math Formula Explorer →