## Arithmetic sequence

Table of Contents

In mathematics, the arithmetic sequence is widely used to find the sequence of the algebraic expression or data observations. It follows a pattern of taking a similar distance between two consecutive numbers. A similar distance can be positive or negative.

Any series that has the same distance among two consecutive numbers from any place refers to the arithmetic sequence. In this post, we will learn the basics of arithmetic sequence along with solved examples.

**What is an arithmetic sequence?**

In algebra, an ordered set of numbers that have a common difference between each consecutive term is known as an arithmetic sequence. In simple words, if the difference between two consecutive terms remains unchanged at any place is said to be the arithmetic sequence.

The list of sequence can be found easily if the first term and the constant term of the sequence is given. You have to add the constant term to the previous term each time to make an accurate arithmetic sequence.

The constant term is also called the common difference; the reason is that the difference of each consecutive term is the same. Natural numbers, whole numbers, integers, even numbers, odd numbers, etc. are examples of an arithmetic sequence.

The arithmetic sequence is of two kinds depending on the constant term.

- Increasing arithmetic sequence
- Decreasing arithmetic sequence

Let’s take a brief overview of the above terms.

**Increasing arithmetic sequence**

In an arithmetic sequence, the terms are in increasing order if the constant term of the sequence is positive. For example, if the initial value of the sequence is 13 and the constant term of the sequence is 4, then the sequence is

13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, ….

The sequence is increasing as the value goes from least to greatest.

**Decreasing arithmetic sequence**

In an arithmetic sequence, the terms are in decreasing order if the constant term of the sequence is negative. For example, if the initial value of the sequence is 39 and the constant term of the sequence is -2, then the sequence is

39, 37, 35, 33, 31, 29, 27, 25, 23, 21, 19, ….

The sequence is decreasing as the value goes from greatest to least.

**Arithmetic sequence formula**

Three kinds of formulas are used in the arithmetic sequence such as:

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

Let’s have a look at these formulas.

**For nth term**

The nth term of the arithmetic sequence can be determined by following the below formula.

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

- x
_{n}= the nth term - x
_{1}= the first term of the sequence - n = total number of terms
- d = common difference

**For the sum of the sequence**

The sum of the sequence can be calculated easily by following the below formula.

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

- x
_{1}= the first term of the sequence - n = total number of terms
- d = common difference

**For finding the common difference **

To find the common difference in the sequence, follow the below formula.

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

**Examples of the arithmetic sequence**

The arithmetic sequence can be determined either by using its formulas. Let us take a few examples to solve the problems of arithmetic sequence manually.

**For the nth term of the sequence**

**Example 1:**

Evaluate the 18^{th} term of the sequence, if the arithmetic sequence is 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, …

**Solution **

**Step I:** Firstly, write the given sequence of numbers.

15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, …

**Step II:** Now evaluate the first term and the constant term of the sequence.

n = 18

a_{1} = 15

a_{2} = 19

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

Common difference = d = 19 – 15

Common difference = d = 4

**Step III:** Now take the formula for finding the nth term of the sequence.

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

**Step IV:** Now place the first term, common term, and total terms in the above formula.

18^{th} term of the sequence= a_{18} = x_{1} + (18 – 1) * d

= 15 + (18 – 1) * 4

= 15 + (17) * 4

= 15 + 68

= 83

Use the nth term calculator to ease up the calculations of finding the nth term and sum of the sequence in a fraction of seconds.

**Example 2**

Evaluate the 139^{th} term of the sequence, if the arithmetic sequence is 116, 121, 126, 131, 136, 141, 146, 151, 156, 161, 166, 171, …

**Solution **

**Step I:** Firstly, write the given sequence of numbers.

116, 121, 126, 131, 136, 141, 146, 151, 156, 161, 166, 171, …

**Step II:** Now evaluate the first term and the constant term of the sequence.

n = 139

a_{1} = 116

a_{2} = 121

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

Common difference = d = 121 – 116

Common difference = d = 5

**Step III:** Now take the formula for finding the nth term of the sequence.

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

**Step IV:** Now place the first term, common term, and total terms in the above formula.

139^{th} term of the sequence= a_{139} = x_{1} + (139 – 1) * d

= 116 + (139 – 1) * 5

= 116 + (138) * 5

= 116 + 690

= 806

**For the sum of the arithmetic sequence**

**Example 3:**

Evaluate the sum of the first 19 terms of the sequence, if the arithmetic sequence is 3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, …

**Solution **

**Step I:** Firstly, write the given sequence of numbers.

3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, …

**Step II:** Now evaluate the first term and the constant term of the sequence.

n = 19

a_{1} = 3

a_{2} = 9

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

Common difference = d = 9 – 3

Common difference = d = 6

**Step III:** Now take the formula for finding the sum of the sequence.

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

**Step IV:** Now place the first term, common term, and total terms in the above formula.

Sum of the sequence = s = 19/2 * (2a_{1} + (19 – 1) * d)

= 19/2 * (2(3) + (19 – 1) * 6)

= 19/2 * (2(3) + (18) * 6)

= 19/2 * (6 + (18) * 6)

= 19/2 * (6 + 108)

= 19/2 * 114

= 9.5 * 114

= 1083

**Conclusion **

In this article, we have covered all the basics of the arithmetic sequence such as definition, kinds, formulas, and solved examples. Now you can easily solve any problem related to arithmetic sequence by following the above post.