Geometric Sequence Explicit Formula Calculator

Calculator Inputs

Known Term Index

Example: 1 means you know a₁.

Known Term Value

This is the value of the known term.

Common Ratio

Each term is multiplied by this ratio.

Target Term Index

Find the value of this term.

Terms to Generate

Used for the results table and graph.

Terms for Finite Sum

Calculates the sum from the known term onward.

Decimal Precision

Controls rounding in displayed answers.

Example Data Table

Example input: known index = 1, known term = 5, common ratio = 3.

n	Explicit Form	Value
1	5 × 3^(1 - 1)	5
2	5 × 3^(2 - 1)	15
3	5 × 3^(3 - 1)	45
4	5 × 3^(4 - 1)	135
5	5 × 3^(5 - 1)	405

Formula Used

Explicit formula: a_n = a_k × r^{(n - k)}

Here, a_k is the known term value, k is its index, r is the common ratio, and n is the target index. This lets you calculate any term directly without listing every earlier term.

Finite sum: S = a_k × (1 - r^m) ÷ (1 - r), when r ≠ 1

Constant ratio case: If r = 1, then every term equals a_k, and the finite sum is m × a_k.

Infinite sum: Sum to infinity exists only when |r| < 1, and equals a_k ÷ (1 - r).

How to Use This Calculator

Enter the index of the known term, such as 1 for a₁.
Enter the value of that known term.
Enter the common ratio between consecutive terms.
Enter the target index you want to calculate.
Choose how many terms to generate for the table and graph.
Set how many terms should be used in the finite sum.
Choose a decimal precision for the displayed output.
Press the calculate button to show results above the form.
Use the CSV and PDF buttons to export the generated results.

Frequently Asked Questions

1. What is a geometric sequence?

A geometric sequence multiplies each term by the same constant ratio to get the next term. Common examples include 2, 6, 18, 54 and 81, 27, 9, 3.

2. What does the explicit formula do?

The explicit formula finds any chosen term directly. You do not need to calculate all earlier terms first, which makes it useful for large indexes.

3. What is the common ratio?

The common ratio is the number used to multiply one term to get the next. For 4, 12, 36, the ratio is 3.

4. Can the ratio be negative?

Yes. A negative ratio creates alternating signs. For example, 5, -10, 20, -40 uses a ratio of -2.

5. What happens when the ratio is 1?

Every term stays the same. The sequence becomes constant, and the finite sum equals the term value multiplied by the number of terms.

6. When does the infinite sum exist?

The infinite sum exists only when the absolute value of the ratio is less than 1. Otherwise, the series does not converge.

7. Why are generated terms shown in a table and graph?

The table helps verify exact values, while the graph shows growth, decay, or alternating behavior visually. Together they make pattern recognition much easier.

8. Can I start from a known term other than a₁?

Yes. This calculator accepts any known index and term value. It then builds the explicit formula around that starting point correctly.