Introduction to Arithmetic Sequence: Definition, Formula, and Examples

In algebra, the term arithmetic sequence is frequently used to determine the sequences of the given data. It follows a way by taking the common difference among two terms to determine the sequence. The sequences can be defined as the ordered group.

Any kind of series of numbers in which the common distance among two consecutive numbers is the same is said to be the sequence and common distance is said to be the constant. In this article, we’ll study the definition and formula of arithmetic sequence along with a lot of examples.

What is an arithmetic sequence?

In algebra, a list of numbers or integers in which the change among two successive terms is constant is said to be the arithmetic sequence. This sequence must have the same constant term among two successive terms, but the first term or starting term can be taken from any number.

The arithmetic sequence can also be defined as a sequence of integers that have a constant difference between two successive numbers. If you have given the first term and the common difference then you can easily make a list of sequences just by adding the constant term to each previous term.

The term constant can also be known as the common difference because the difference among each consecutive term is the same. There are many examples of arithmetic sequences such as the set of integers, set of natural numbers, set of whole numbers, set of even numbers, etc.

In each well-known set of the number system, the common difference among two consecutive terms is the same so these sets are said to be the arithmetic sequences. In algebra, the arithmetic sequence can be increasing or decreasing depending on the common difference.

The sequence is increasing if the common difference is a positive number. While the sequence is said to be decreasing if the common difference is a negative number. For example, if the starting value of the sequence is 32 and the common difference is 2 then the sequence must be:

32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, …

The common difference among the numbers is positive. So, the sequence is said to be increasing. For decreasing sequence, let the first term of the sequence is 10 and the common difference among the numbers is -3, then the sequence must be:

10, 7, 4, 1, -2, -5, -8, -11, -14, -17, -20, …

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Arithmetic sequence formula

There are three kinds of formulas for arithmetic sequence i.e., for the nth term, for a sum of sequences, and the common difference.

  1. The formula for determining the nth term of the sequence is:

nth term of the sequence= an = a1 + (n – 1) * d

  • The formula for finding the sum of the sequence is:

Sum of the sequence = s = n/2 * (2a1 + (n – 1) * d)

  • The formula for finding the constant of two consecutive terms is:

Common difference = d = an – an-1

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How to determine the arithmetic sequence?

The arithmetic sequence can be determined either by using its formulas or an arithmetic sequence calculator. By using a calculator, you can get the result in a fraction of second with steps. Let us take a few examples to solve the problems of arithmetic sequence manually.

Example-1: For the nth term of the sequence

Determine the 14th term of the sequence, if the arithmetic sequence is 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, …

Solution

Step I: First of all, take the given sequence of numbers.

5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, …

Step II: Now identify the first term of the sequence and find the common difference of the numbers.

n = 14

a1 = 5

a2 = 8

Common difference = d = a2 – a1

                                           = 8 – 5

                                           = 3

Step III: Now write the formula for the nth term of the arithmetic sequence.

nth term of the sequence= an = a1 + (n – 1) * d

Step IV: Now substitute the n = 14, common difference, and th starting value of the sequence.

14th term of the sequence= a14 = a1 + (14 – 1) * d

                                                       = 5 + (14 – 1) * 3

                                                       = 5 + (13) * 3

                                                       = 5 + 39

                                                       = 44

Example 2

Determine the 115th term of the sequence, if the arithmetic sequence is 15, 22, 29, 36, 43, 50, 57, 64, 71, 79, 86, 93, …

Solution

Step I: First of all, take the given sequence of numbers.

15, 22, 29, 36, 43, 50, 57, 64, 71, 79, 86, 93, …

Step II: Now identify the first term of the sequence and find the common difference of the numbers.

n = 115

a1 = 15

a2 = 22

Common difference = d = a2 – a1

                                            = 22 – 15

                                            = 7

Step III: Now write the formula for the nth term of the arithmetic sequence.

nth term of the sequence= an = a1 + (n – 1) * d

Step IV: Now substitute the n = 115, common difference, and the starting value of the sequence.

115th term of the sequence= a115 = a1 + (115 – 1) * d

                                                            = 15 + (115 – 1) * 7

                                                            = 15 + (114) * 7

                                                            = 15 + 798

                                                            = 813

Example 3: For sum of the arithmetic sequence

Determine the sum of the first 14 terms of the sequence, if the arithmetic sequence is 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, …

Solution

Step I: First of all, take the given sequence of numbers.

4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, …

Step II: Now identify the first term of the sequence and find the common difference of the numbers.

n = 14

a1 = 4

a2 = 7

Common difference = d = a2 – a1

                                           = 7 – 4

                                           = 3

Step III: Now write the formula for the sum of the sequence term of the arithmetic sequence.

Sum of the sequence = s = n/2 * (2a1 + (n – 1) * d)

Step IV: Now substitute the n = 14, common difference, and the starting value of the sequence.

Sum of the sequence = s = n/2 * (2a1 + (n – 1) * d)

                                             = 14/2 * (2a1 + (14 – 1) * d)

                                             = 14/2 * (2(4) + (14 – 1) * 3)

                                             = 14/2 * (2(4) + (13) * 3)

                                             = 14/2 * (8 + (13) * 3)

                                             = 14/2 * (8 + 39)

                                             = 14/2 * (47)

                                             = 7 * (47)

                                             = 329

Summary

In this post, we have learned all the basics of the arithmetic sequence along with formulas and examples. Now after reading the above post, you can easily find the nth term of the sequence and the sum of the sequence by taking a common difference.

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