## Arithmetic Progression: An introduction with definition and examples

**Arithmetic Progression: An introduction with definition and examples**

In set theory and algebra, the arithmetic sequence (progression) is widely used to calculate the sequence of the terms. Algebra is a branch of mathematics that deals with the letters and symbols to denote the numbers in the form of equations.

While the set theory is the collection of objects that are well determined. In this post, we are going to explain the term arithmetic sequence along with its definition, expressions, introduction, and solved examples.

## What is the Arithmetic Progression?

In algebra, the term arithmetic progression which is also known as the arithmetic sequence is a sequence of numbers or integers that have the same common difference (differ from one another by common difference).

For example, the sequence 13, 26, 39, 52, 65, 78, 91, …., is an arithmetic sequence as the common difference between each consecutive term is 13. The sequence of the numbers can be created by taking the initial term of the sequence, a number of terms, and common difference.

The common difference is defined as the difference between two consecutive terms. It is denoted by “d”. such as if the initial term of the sequence is 12 and the common difference among each term is 4, then the sequence will be:

12, 16, 20, 24, 28, 32, …

The sequence is of two types such as:

- Increasing
- Decreasing

The increasing and decreasing sequences are depending on the common difference of the sequences. Here is a brief introduction to these types of an arithmetic sequences.

### 1. Increasing sequence

In the sequence in which the constant term (common difference) is positive then that sequence must be increasing. By adding the positive common difference to each next term the numbers must go from least to greatest.

For example, calculate the sequence if the initial term is 13 and the common difference is 2

13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, …

Hence the sequence is said to be increasing as the sequence is going from least to greatest.

### 2. Decreasing sequence

In the sequence in which the constant term (common difference) is negative then that sequence must be decreasing. By adding the negative common difference to each next term the numbers must go from greatest to least.

For example, calculate the sequence if the initial term is 13 and the common difference is -2

13, 11, 9, 7, 5, 3, 1, -1, -3, -5, -7, …

Hence the sequence is said to be decreasing as the sequence is going from greatest to least.

## Arithmetic Progression’s formulas

Here are three commonly used formulas in arithmetic progression for solving its problems.

- For nth term
- For the sum of the sequence
- For finding the common difference

These formulas are very essential for calculating the sequence and sum of the sequence easily.

### a. For the nth term of the sequence

Here is the general expression for calculating the nth term of the sequence.

**nth term of the sequence= f _{n} = f_{1} + (n – 1) * d**

where f_{n} is the nth term of the sequence, f_{1} is the starting term of the sequence, n is the total number of terms, and d is a common difference.

### b. For the sum of the sequence

Here is the general expression for calculating the sum of the sequence.

**Sum of the sequence = s = n/2 * (2f _{1} + (n – 1) * d)**

Where f_{1} is the starting term of the sequence, n is the total number of terms, and d is a common difference.

### c. For finding the common difference

Here is the general expression for calculating the common difference of the sequence.

**Common difference = d = f _{n} – f_{n-1 }**

An arithmetic sequence calculator can be used to calculate the nth term of the sequence and the sum of the sequence in a fraction of a second along with steps according to the above formulas.

## How to calculate the arithmetic progression?

The arithmetic sequence can be calculated easily by using the formulas of the nth term, common difference, and the sum of the sequence. Below are a few examples of the arithmetic sequence to learn how to calculate it.

**Example 1: For the nth term**

Calculate the 25^{th }term of the sequence by taking the values from the given sequence,

15, 19, 23, 27, 31, 35, 39, 43, 47, 51, …

**Solution **

**Step 1:** First of all, take the given sequence and determine the initial term and the common difference of the sequence.

15, 19, 23, 27, 31, 35, 39, 43, 47, 51, …

Initial term = f_{1} = 15

Second term = f_{2} = 19

Common difference = d = f_{2} – f_{1 }

Common difference = d = 19 – 15

Common difference = d = 4

We have to calculate the 25^{th }term of the sequence so n = 25

**Step 2:** Now take the general formula for calculating the nth term of the sequence.

nth term of the sequence= f_{n} = f_{1} + (n – 1) * d

**Step 3:** Now substitute the common difference (d), the initial term (f_{1}), and n = 25 to the given formula.

25^{th} term of the sequence= a_{21} = 15 + (25 – 1) * 4

25^{th} term of the sequence= a_{21} = 15 + (24) * 4

25^{th} term of the sequence= a_{21} = 15 + 96

25^{th} term of the sequence= a_{21} = 111

**Example 2: For the sum of the sequence**

Calculate the sum of the first 30 terms of the sequence by taking the values from the given sequence,

21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, …

**Solution **

**Step 1:** First of all, take the given sequence and determine the initial term and the common difference of the sequence.

21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, …

Initial term = f_{1} = 21

Second term = f_{2} = 27

Common difference = d = f_{2} – f_{1 }

Common difference = d = 27 – 21

Common difference = d = 6

We have to calculate the sum of the first 30 terms of the sequence so n = 30

**Step 2:** Now take the general formula for calculating the sum of the sequence.

Sum of the sequence = s = n/2 * (2f_{1} + (n – 1) * d)

**Step 3:** Now substitute the common difference (d), the initial term (f_{1}), and n = 30 to the given formula.

Sum of the first 30 terms = s = 30/2 * (2(21) + (30 – 1) * 6)

Sum of the first 30 terms = s = 30/2 * (42 + (29) * 6)

Sum of the first 30 terms = s = 30/2 * (42 + 29 * 6)

Sum of the first 30 terms = s = 30/2 * (42 + 174)

Sum of the first 30 terms = s = 30/2 * (216)

Sum of the first 30 terms = s = 30/2 * 216

Sum of the first 30 terms = s = 15 * 216

Sum of the first 30 terms = s = 3240

## Wrap up

In this post, we have covered all the basics of this technique of finding the nth term and sum of the sequence. Now you can grab all the basics of arithmetic progression from this post by learning it. Once you grab the basics of this topic, you will master it.