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Geometric Sequence Calculator

Select and enter the values to calculate the geometric progression and related parameters in the sequence.

geometric

Find:

Inputs (a₁)

Number of Terms (r)

Common Ratio (n)

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Geometric Sequence Calculator

The geometric sequence calculator helps you determine missing terms, the common ratio, and the sum of a geometric series with clear step-by-step explanations. It can calculate:

First term (a₁)
Common ratio (r)
nᵗʰ term of the sequence
Sum of the first n terms

Note: This calculator works with real numbers only and does not support complex ratios or infinite series calculations directly.

How to Use the Geometric Sequence Calculator

Choose the value you want to find.
Enter the known values into the input boxes.
Click Calculate.
Instantly view the computed result along with steps.

What is a Geometric Sequence?

“A geometric sequence is a number pattern in which each term is obtained by multiplying the previous term by a fixed, non-zero constant called the common ratio (r).”

If you need to find a previous term, simply divide the known term by the common ratio.

Formula for the nᵗʰ Term

\( a_n = a_1 \cdot r^{\,n-1} \)

Where:

aₙ = value of the nth term
a₁ = first term
r = common ratio
n = term position

Main Components

First Term (a₁): The starting number of the sequence.
Common Ratio (r): The multiplier between consecutive terms.
nᵗʰ Term: The value located at position n.

How to Calculate Terms in a Geometric Sequence

To determine any term, raise the common ratio to the power of (n − 1) and multiply the result by the first term.

Example

Consider the sequence: 5, 15, 45, 135, …

Given:

a₁ = 5
r = 3
n = 4

Find the 4th term:

\( a_4 = 5 \cdot 3^{4-1} = 5 \cdot 3^3 = 5 \cdot 27 = 135 \)

Sum of the first 4 terms:

\( S_n = a_1 \frac{1 - r^n}{1 - r} \quad (r \ne 1) \)

\( S_4 = 5 \frac{1 - 3^4}{1 - 3} = 5 \frac{1 - 81}{-2} = 5 \cdot 40 = 200 \)

Common Questions

Types of Geometric Sequences

Finite Geometric Sequence: Contains a specific number of terms.

\( S_n = \frac{a_1 (1 - r^n)}{1 - r} \)

Infinite Geometric Sequence: Continues indefinitely and converges only when |r| < 1.

\( S_\infty = \frac{a_1}{1 - r} \)

Practical Applications

Geometric sequences are widely used in:

Compound interest calculations
Population and bacterial growth models
Computer science (exponential algorithms)
Financial forecasting
Radioactive decay models
Signal processing and amplification

Steps to Find the Sum of a Geometric Series

Calculate \( r^n \).
Subtract this value from 1.
Divide by (1 − r).
Multiply the result by the first term a₁.

Finding the Common Ratio (r)

Divide any term by the term before it:

\( r = \frac{a_{n+1}}{a_n} \)

Finding the nᵗʰ Term

Compute \( r^{n-1} \).
Multiply by the first term a₁.

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