# Geometric Sequence Calculator

## Solve geometric progressions step by step

The calculator will find the terms, common ratio, sum of the first $n$ terms and, if possible, the infinite sum of the geometric sequence from the given data, with steps shown.

Related calculator: Arithmetic Sequence Calculator

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$S_{n}$ is the sum of the first $n$ terms.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

The Geometric Sequence Calculator is your trusted companion for identifying a specific term or computing the full geometric sequence using your provided inputs. Our calculator handles the task in moments and with a few simple clicks, ensuring you receive the correct output immediately.

## How to Use the Geometric Sequence Calculator?

• ### Input

Begin by inputting the known data. This may include the first term of the sequence, the common ratio, and, if applicable, the specific term number you are seeking.

• ### Calculation

Once you've entered the necessary data, click the "Calculate" button.

• ### Result

The calculator will promptly display either the desired term or the full sequence based on your inputs. Make sure to cross-check the output for accuracy.

## What Is a Geometric Sequence?

A geometric sequence is a particular kind of number series that has a consistent pattern facilitated by a fixed number known as the ratio. In essence, starting from the first term, every subsequent term in the series is the product of the previous term and this fixed ratio.

The formula for the nth term of a geometric sequence is:

$$a_n=a_1r^{n-1}$$

Where:

• $a_n$ is the nth term of the sequence.
• $a_1$ represents the first term of the sequence.
• $r$ is the common ratio, which is the fixed number we multiply by to get the next term.
• $n$ indicates the position of the term in the sequence.

Breaking Down the Components:

• First Term $\left(a_1\right)$: This is the starting point of the geometric sequence. All subsequent terms are determined based on this value and the common ratio.
• Common Ratio $\left(r\right)$: This is the factor by which we multiply a term to get the next term in the sequence. It's essential to note that $r$ is non-zero because a zero value would make the entire sequence zero after the first term.

For example, given the first term $a_1=2$ and the common ratio $r=3$ the geometric sequence would be represented as:

$$2,6,18,54,162,\ldots$$

Here, the 2nd term is $2\cdot3=6$, the 3rd term is $6\cdot3=18$ and so on.

## Why Choose Our Quadratic Equation Calculator?

• ### Accuracy and Precision

Our calculator has been carefully developed and tested to ensure that the components of a geometric sequence are computed with the highest level of accuracy every time.

• ### User-Friendly Interface

With a simple and intuitive design, users can easily input values and understand results regardless of their mathematical background.

• ### Speed

Time is of the essence. Our calculator processes your input instantly, delivering the result without delay.

• ### Detailed Solutions

Besides providing the answer, our calculator also offers step-by-step solutions for those keen on understanding the method behind the results.

### FAQ

#### What's the difference between an arithmetic and geometric sequence?

While both are sequences, the main difference lies in their patterns. In an arithmetic sequence, the difference between consecutive terms is constant. In a geometric sequence, each term is a product of the previous term and a constant ratio.

#### What is a geometric sequence?

A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the ratio.

#### What is the Geometric Sequence Calculator used for?

The Geometric Sequence Calculator is designed to help users find a specific term or calculate the entire geometric sequence based on given values such as the first term and common ratio.

#### Does the calculator handle geometric series?

While the primary function is to find terms in a geometric sequence, the calculator also offers the ability to compute the sum of a geometric series.