An arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. Use our handy calculator tool to define the numbers and compute the sequence.

## What is a sequence?

We know that a sequence is a function whose domain is the set N of natural numbers. Moreover, a sequence whose range is a subset of R is called a real sequence.In other words, a real sequence is a function with domain N and the range a subset of the set R of real numbers. There are several ways of representing a real sequence. One way to represent a real sequence is to list its first few terms till the rule for writing down other terms becomes clear, for example, 1, 3, 5 ….. is a sequence whose nth term is ( 2n – 1 ).

Another way of representing a real sequence is to give a rule of writing the nth term of a sequence. For example, the sequence, 1, 3, 5, 7 ….. can be written as a_{n} = 2n – 1.

Sometime we represent a real sequence by using a recursive relation. For example, the Fibonacci sequence is given by

a_{1} = 1, a_{2} = 1 and a_{n + 1} = a_{n} + a_{n – 1}, n ≥ 2

The terms of this sequence 1, 1, 2, 3, 5, 8, ……..

## What is an arithmetic sequence?

**An arithmetic sequence is a sequence of numbers in order in which the difference of any two consecutive numbers is a constant value. It is also known as arithmetic progression (AP). ** There are three main terms associated with an arithmetic sequence –

- Common difference ( d ) which is calculated as d = a
_{2}– a_{1}= a_{3}– a_{2}= ……. = a_{n}– a_{n – 1} - the nth term of the arithmetic sequence
- Sum of first n terms ( S
_{n})

The arithmetic sequence can also be written in terms of common difference, as follows –

a, a + d, a + 2d, a + 3d, a + 4d, ………. ,a + (n – 1) d

where “ a “ is the first term of the sequence.

### How to find the nth term of an arithmetic sequence?

The formula for finding the nth term of an arithmetic sequence is given by –

**a _{n} = a + (n − 1) × d. where**

a = First term

d = Common difference

n = number of terms

a_{n} = nth term

Let us understand it through an example.

**Example**

Suppose we wish to find the nth term of a sequence 1, 2, 3, 4, 5, ……………. a_{n} and there are 15 terms in the sequence.

** Solution**

We have been given a sequence 1, 2, 3, 4, 5, ……………. a_{n} which has 15 terms and we are required to find the nth term of this sequence. Let us summarise this information in mathematical terms. We have,

Number of terms, n = 15

First term, a = 1

Common difference, d = 2 – 1

Now, we know that finding the nth term of an arithmetic sequence is given by** a _{n} = a + (n − 1) × d. Substituting the given values in this equation, we have,**

**a _{n} = 1 + ( 15 – 1 ) x 1**

**⇒**** a _{n} = 1 + 14 x 1**

**⇒**** a _{n} = 1 + 14**

**⇒ a _{n} = 15**

## How arithmetic sequence is calculated using the nth term?

How can we obtain the arithmetic sequence if we have been given the nth value of the sequence and the common difference between the terms? Let us understand it using an example.

Suppose we have been given that the 5^{th} term of a sequence is 20. We have also been given that the common difference between the terms is 2. Using this information, we need to find the first 10 terms of this sequence. How will we find out?

We can see that the 5^{th} term of the required arithmetic sequence is 20. We have also been given that the common difference between the terms is 2. This means that the next term after 20 will be 20 + 2 = 22, which will be considered as the sixth term. Similarly, going backwards from 20, we will have the first four terms of the sequence as 12, 14, 16 and 18. This means that the first 10 terms of the resultant arithmetic sequence will be 12, 14, 16, 18, 20, 22, 24, 26, 28 and 30. This is how we can generate the arithmetic sequence if we have been given the nth value of the sequence and the common difference between the terms.

## How to use an arithmetic sequence calculator?

We have just learnthow to generate the arithmetic sequence if we have been given the nth value of the sequence and the common difference between the terms. Let us now see how to perform the same calculations using the arithmetic sequence generator. We shall use the same example that we considered above where we were given that the 5^{th} term of a sequence is 20. We have also been given that the common difference between the terms is 2. The following steps will be used for this purpose –

**Step 1 – **The first step is to enter the value of n. since we have been given the 5^{th} term of the sequence, therefore, we will enter n = 5 in the box given. Below is a snapshot of what our entry would look like –

**Step 2** – The next step is to enter the nth value corresponding to the term n that we entered in the first step. Since we have been given that the 5^{th} term of a sequence is 20, therefore, in the nth value of the arithmetic sequence calculator, we will enter 20. Below is a snapshot of what our entry would look like –

**Step 3** – the third step is to enter the common difference. We have been given that the common difference between the terms is 2, therefore, we will enter 2 against the common difference box in the arithmetic sequence calculator. . Below is a snapshot of what our entry would look like –

**Step 4** – The fourth step is to enter the number of terms of the arithmetic sequence that we want to obtain. Since we are required to find the first 10 terms, we will enter 1 in the total number of terms box in the arithmetic sequence calculator. . Below is a snapshot of what our entry would look like –

**Step 5** – Now that we are done with entering all the required information, the last step is to find the arithmetic sequence which can be done just by clicking on the calculate button. Below is a snapshot of how the result would be displayed by the arithmetic sequence calculator.

From above we can clearly see that not only is the sequence displayed, but also along with it we get to know the formula and the logic / calculation behind the generation of each term of the sequence. This means that our arithmetic sequence calculator not just helps you check the final result, but it also assists in the learning process where you can go through the corresponding formulas and the steps involved in obtaining the desired result. Repeated use of the arithmetic sequence calculator will certainly help you have a grasp on the various aspects of finding various arithmetic sequences and it also helps in the visualisation of the generation of terms in an arithmetic sequence.