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Arithmetic sequence: An introduction with examples

Arithmetic sequence: An introduction with examples

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In mathematics, the arithmetic sequence is widely used to find the correct sequence of the given data values. The sequence means the ordered list of values in a particular pattern like taking a common difference.

The list of numbers that have the same common difference is said to be the sequence with accurate order. There are several ways to determine the sequences of the list of numbers. In this article, we will learn all the basics of the arithmetic sequence along with solved examples.

What is the arithmetic sequence?

A list of numbers in which the difference among two successive terms is the same or constant is said to be the arithmetic sequence. The sequence of arithmetic can be started from any number either positive or negative but the common difference between the terms must always be the same.

Integers, whole numbers, natural numbers, odd numbers, even numbers, etc. are examples of the arithmetic sequence as the difference between each successive term of these numbers is the same.

The positive difference among the sequences is said to be the increasing sequence or the positive difference for the positive integers is said to be the increasing sequence. 

For example, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, … is an increasing arithmetic sequence as the starting term is 35 and the common difference among two successive terms is two. While negative difference among two consecutive terms or a positive difference between two terms for the negative integer is said to be the decreasing sequence.

For example, 11, 8, 5, 2, -1, -4, -9, -10, -13, -16, -19, … is a decreasing sequence as the starting term is 11 and the difference between two successive terms is negative three.

Arithmetic sequence formula

The formulas for the arithmetic sequence are of three kinds such as for the nth term, the sum of series, & the common difference. 

The formula for finding the nth term of the sequence is: 

nth term = bn = b1 + (n – 1) * d

The formula for determining the sum of the sequence is:

Sum of the sequence = s = n/2 * (2b1 + (n – 1) * d)

The formula for finding the difference between two consecutive terms is: 

Common difference = d = bn – bn-1 

How to find the arithmetic sequence?

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

Example-I: For the nth term 

Calculate the 17th term of the sequence, if the arithmetic sequence is 2, 7, 12, 17, 22, 27, 32, 37, 42, 47, 52, 57, …

Solution 

Step-I: First of all, write the list of given numbers.  

2, 7, 12, 17, 22, 27, 32, 37, 42, 47, 52, 57, …

Step-II: Identify the first term, nth term, and the difference between two consecutive terms.  

nth term = 17

first term = b1 = 2

second term = b2 = 7

Common difference = d = b2 – b

                                           = 7 – 2

                                           = 5

Step-III: Take the formula of the nth term to calculate it. 

nth term of the sequence= bn = b1 + (n – 1) * d

Step-IV: Now put the common difference, first term, and the nth term of the sequence in the formula.

17th term of the sequence= b17 = b1 + (17 – 1) * d

                                                        = 2 + (17 – 1) * 5 

                                                        = 2 + (16) * 5 

                                                        = 2 + 80

                                                        = 82

Use an arithmetic sequence calculator to find the nth term of the sequence in a fraction of seconds with steps.

Example II

Calculate the 150th term of the sequence, if the arithmetic sequence is 1, 8, 15, 22, 29, 36, 43, 50, 57, 64, 71, 78, …

Solution 

Step-I: First of all, write the list of given numbers.  

1, 8, 15, 22, 29, 36, 43, 50, 57, 64, 71, 78, …

Step-II: Identify the first term, nth term, and the difference between two consecutive terms.  

nth term = 150

first term = b1 = 1

second term = b2 = 8

Common difference = d = b2 – b

                                           = 8 – 1

                                           = 7

Step-III: Take the formula of the nth term to calculate it. 

nth term of the sequence= bn = b1 + (n – 1) * d

Step-IV: Now put the common difference, first term, and the nth term of the sequence in the formula.

150th term of the sequence= b150 = b1 + (150 – 1) * d

                                                            = 1 + (150 – 1) * 7

                                                            = 1 + (149) * 7

                                                            = 1 + 1043

                                                            = 1044

Example III: For sum of the arithmetic sequence

Determine the sum of the first 20 terms of the sequence, if the arithmetic sequence is 3, 8, 13, 18, 23, 28, 33, 38, 43, 48, 53, 58, …

Solution 

Step-I: First of all, write the list of given numbers.  

3, 8, 13, 18, 23, 28, 33, 38, 43, 48, 53, 58, …

Step-II: Identify the first term, nth term, and the difference between two consecutive terms.  

nth term = 20

first term = b1 = 3

second term = b2 = 8

Common difference = d = b2 – b

                                            = 8 – 3

                                            = 5

Step-III: Take the formula of the Sum of the sequence to calculate it.

Sum of the sequence = s = n/2 * (2b1 + (n – 1) * d)

Step-IV: Now put the common difference, first term, and the nth term of the sequence in the formula.

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

                                             = 20/2 * (2b1 + (20 – 1) * d)

                                             = 20/2 * (2(3) + (20 – 1) * 5)

                                             = 20/2 * (2(3) + (19) * 5)

                                             = 20/2 * (6 + (19) * 5)

                                             = 20/2 * (6 + 95)

                                             = 20/2 * 101

                                             = 10 * 101

                                             = 111

Summary 

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

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