An Introduction to Arithmetic Sequence: Defined and Explained with Examples

The arithmetic sequence is a well-known type of sequence in which the difference between two consecutive terms remains the same. A collection of numbers that follows a pattern is said to be a sequence.

The sequences are of two types such as arithmetic sequence and geometric sequence. There are two different ways to evaluate the sequences. In this post, we’ll describe the arithmetic sequence, its formulas, and examples.

What is the arithmetic sequence?

The arithmetic sequence is a sequence in which the common differences between every consecutive term are the same. Such as the sequence 3, 7, 11, 15, 19, 23, 27… is an arithmetic sequence as the common difference between every successive term is 4.

It can also be defined as the next term obtained by adding the fixed common difference to the previous term. Such as if the starting term of the sequence is 2 and the common difference is 6 then the arithmetic sequence is 2, 8, 14, 20, 26, 32, 38, 44 …

The common difference can be positive or negative. The sequence with the positive common difference is said to be the increasing arithmetic sequence. Such as the first term of the sequence is 11 and the common difference is 3 then the increasing common difference is:

11, 1, 17, 20, 23, 26, 29, 32, 35 …

The sequence with the negative common difference is said to be the decreasing arithmetic sequence. Such as the first term of the sequence is 41 and the common difference is -2 then the decreasing common difference is:

41, 39, 37, 35, 33, 31, 29, 27, 25, 23 …

Formulas of Arithmetic Sequence

The formulas of the arithmetic sequence are:

How to evaluate the sum of the sequence and the nth term of the sequence?

The formulas for the nth term of the sequence and the sum of the sequence are helpful for evaluating the sum and nth term of the sequence. Here are examples for understanding how to evaluate them.

For finding the nth term

Example 1

Find the 33^rd term of the given sequence with the help of the arithmetic sequence for the nth term.

1, 10, 19, 28, 37, 46, 55, 64, 73…

Solution

Step 1: First of all, take the given sequence and find the common difference of the sequence by taking the difference between two consecutive terms.

1, 10, 19, 28, 37, 46, 55, 64, 73…

1st term = a₁ = 1

2nd term = a₂ = 10

Common difference = d = a₂ – a₁

Common difference = d = 10 – 1

Common difference = d = 9

The n value will be 33 as we have to find the 33^rd term of the sequence.

Step 2: Take the general formula for finding the nth term of the sequence.

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

Step 3: Now place the n = 33, the first term of the sequence, and the common difference to the above formula.

33^rd term of the sequence = a₃₃ = 1 + (33 – 1) * 9

33^rd term of the sequence = a₃₃ = 1 + (32) * 9

33^rd term of the sequence = a₃₃ = 1 + (288)

33^rd term of the sequence = a₃₃ = 1 + (288)

33^rd term of the sequence = a₃₃ = 289

Example 2

Find the 15^th term of the given sequence with the help of the arithmetic sequence for the nth term.

2, 7, 12, 17, 22, 27, 32, 37, 42…

Solution

Step 1: First of all, take the given sequence and find the common difference of the sequence by taking the difference between two consecutive terms.

2, 7, 12, 17, 22, 27, 32, 37, 42…

1st term = a₁ = 2

2nd term = a₂ = 7

Common difference = d = a₂ – a₁

Common difference = d = 7 – 2

Common difference = d = 5

The n value will be 15 as we have to find the 15^th term of the sequence.

Step 2: Take the general formula for finding the nth term of the sequence.

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

Step 3: Now place the n = 15, the first term of the sequence, and the common difference to the above formula.

15^th term of the sequence = a₁₅ = 1 + (15 – 1) * 5

15^th term of the sequence = a₁₅ = 1 + (14) * 5

15^th term of the sequence = a₁₅ = 1 + (70)

15^th term of the sequence = a₁₅ = 1 + 70

15^th term of the sequence = a₁₅ = 71

The nth term of the sequence can also be determined with the help of online tools.

Solve through https://www.allmath.com/arithmetic-progression.php

For finding the sum of the sequence

Example

Determine the sum of the first 20 terms of the given sequence.

9, 11, 13, 15, 17, 19, 21, 23, 25, …

Solution

Step 1: First of all, take the given sequence and find the common difference of the sequence by taking the difference between two consecutive terms.

9, 11, 13, 15, 17, 19, 21, 23, 25, …

Initial term = a₁ = 9

Second term = a₂ = 11

Common difference = d = a₂ – a₁

Common difference = d = 11 – 9

Common difference = d = 2

The n will be 20 as we have to find the sum of the first 20 terms.

Step 2: Take the general formula for finding the sum of the sequence.

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

Step 3: Substitute the 1^st term, common difference, and the total number of n values to the above formula of the finding Sum of the sequence.

Sum of the first 20 terms = s = 20/2 * (2(9) + (20 – 1) * 2)

Sum of the first 20 terms = s = 20/2 * (2(9) + (19) * 2)

Sum of the first 20 terms = s = 20/2 * (18 + 19 * 2)

Sum of the first 20 terms = s = 20/2 * (18 + 38)

Sum of the first 20 terms = s = 20/2 * 56

Sum of the first 20 terms = s = 10 * 56

Sum of the first 20 terms = s = 560

Wrap Up

The arithmetic sequence is a well-known sequence in which the sequence is written in a pattern. The common difference between two consecutive terms is fixed. The formulas of the nth term and the sum of the sequence are helpful for solving the problems of an arithmetic sequence