# Arithmetic Sequence Calculator

## Solve arithmetic progressions step by step

The calculator will find the terms, common difference and sum of the first $n$ terms of the arithmetic sequence from the given data, with steps shown.

Related calculator: Geometric 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.

Your arithmetic sequence calculations just got easier with our Arithmetic Sequences Calculator. Ideally suited for understanding and calculating arithmetic sequences, this tool helps you quickly discover the nth term or compute the sum of an arithmetic sequence. Make arithmetic sequence calculations a breeze with our user-friendly online calculator.

## How to Use the Arithmetic Sequence Calculator?

• ### Input

Enter the information you have: term numbers (positions of the terms in the sequence), term values, common difference, sum values, etc.

• ### Calculation

After inputting all the information, click on the "Calculate" button.

• ### Result

The calculator will immediately provide the desired result.

## What Is an Arithmetic Sequence?

An arithmetic sequence, also known as an arithmetic progression, is a sequence of numbers in which the difference between two consecutive terms is constant. This difference is typically referred to as the "common difference."

An arithmetic sequence is usually written in the form:

$$a_1,a_1+d,a_1+2d,a_1+3d,a_1+4d,\ldots,$$

where $a_1$ is the first term, and $d$ is the common difference between the terms.

Example of an Arithmetic Sequence

For instance, consider an arithmetic sequence $3,7,11,15,\ldots$. In this sequence, $a_1$ (the first term) is $3$, and $d$ (the common difference) is $4$. You can see that each term in the sequence is $4$ more than the previous term.

The nth Term of an Arithmetic Sequence

The general formula to find the nth term of an arithmetic sequence is:

$$a_n=a_1+d(n-1)$$

Here, $a_n$ denotes the nth term, $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number.

Using the above example, if we want to find the 5th term $\left(n=5\right)$, we substitute these values into the formula:

$$a_5=3+4\cdot(5-1)=3+16=19$$

So the 5th term of the sequence is $19$.

The Sum of an Arithmetic Sequence

The sum $S_n$ of the first $n$ terms of an arithmetic sequence can be calculated using the formula:

$$S_n=\frac{(a_1+a_n)n}{2}=\frac{(2a_1+d(n-1))n}{2}$$

Here, $a_n$ is the nth term. For the above sequence, if we want to find the sum of the first $5$ terms:

$$S_5 =\frac{(2\cdot3+4\cdot(5-1))\cdot5}{2}=\frac{110}{2}=55$$

So the sum of the first $5$ terms of the sequence is $55$.

Understanding these principles of arithmetic sequences can help you use the Arithmetic Sequence Calculator more effectively.

## What Is the Difference Between a Sequence and a Series in Mathematics?

A sequence and a series are two foundational concepts in mathematics, specifically within calculus and analysis, but they serve different purposes and concepts.

Sequence

A sequence is a list of numbers (or other elements such as functions or sets) arranged in a specific order. Each number in a sequence is called a term. Sequences can be finite or infinite. For example, the sequence of natural numbers $\left\{1,2,3,4,5,\ldots\right\}$ is an infinite sequence.

Series

A series, on the other hand, is the sum of a sequence of numbers. If a sequence is a list, a series is the addition of those listed elements. A series can be represented as the sum of the terms of a sequence. For example, given the sequence $\left\{1,2,3,4,5,\ldots\right\}$, the corresponding series would be $1+2+3+4+5+\ldots$.

The key difference is that a sequence is a list of elements, while a series is a sum of elements.

## Why Choose Our Arithmetic Sequence Calculator?

• ### Efficiency

Our calculator performs quick and accurate calculations, saving you significant time. Whether you're finding the nth term or the sum of an arithmetic sequence, our tool delivers results promptly.

• ### User-Friendly Interface

With a simple and intuitive design, our Arithmetic Sequence Calculator is incredibly easy to use. You'll have no trouble navigating through the interface, even if you're a first-time user.

• ### Educational Value

The calculator is not just about delivering results. It also helps understand the underlying arithmetic sequence concept, showing all the steps involved in the calculations. This makes it an excellent learning tool for students.

• ### Versatility

This tool is not just for students; professionals can also use it for quick calculations. It's a versatile tool for anyone dealing with arithmetic sequences.

• ### Accessibility

Our Arithmetic Sequence Calculator is online and free to use, making it accessible anytime, anywhere.

### FAQ

#### How do I tell if a sequence is arithmetic?

You can determine whether a sequence is arithmetic by checking the difference between successive terms. If the difference is constant, the sequence is arithmetic. For example, in the sequence $2,4,6,8$, the difference between each term is $2$, making it an arithmetic sequence.

#### What is an arithmetic sequence?

An arithmetic sequence is a sequence of numbers where the difference between successive terms is constant. This difference is often referred to as the "common difference."

#### Can the calculator calculate the sum of an arithmetic sequence?

Yes, our Arithmetic Sequence Calculator can calculate the sum of an arithmetic sequence. The formula for calculating the sum of the first $n$ terms is $S_n=\frac{(a_1+a_n)n}{2}$, where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, and $a_n$ is the nth term.

#### Is the Arithmetic Sequence Calculator free to use?

Absolutely! Our Arithmetic Sequence Calculator is completely free to use. It's accessible online, so you can use it anytime, anywhere.