An arithmetic sequence is a mathematical expression consisting of a sequence of numbers in which each number is the same amount larger than the number before it. It is the best technique for quickly understanding patterns in data and can be used to find sums of numbers or calculate future values in a sequence.

In this blog post, we will take a look at the definition, formula, and examples of an arithmetic sequence.

What is an arithmetic sequence?

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is always the same. This fixed difference is called the common difference. For example, the sequence 5, 9, 13, 17 is an arithmetic sequence because the common difference is 4 (9 – 5 = 4, 13 – 9 = 4, 17 – 9 = 4).

Arithmetic sequences can be written in form a, a + d, a + 2d, a + 3d… where “a” is the first term in the sequence and d is a common difference. For example, the sequence 3, 7, 11, 15 can be written as 3, 3+4, 3+8, 3+12, …

In arithmetic sequences, the nth term and the sum of the terms of an arithmetic series (a finite arithmetic sequence) can be calculated using formulas.

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

Orders of an arithmetic sequence

The arithmetic sequence can be divided into two orders such as

1. Increasing Arithmetic Sequence
2. Decreasing Arithmetic Sequence

An increasing arithmetic sequence is a sequence in which the terms of the sequence increase as the index of the term increases. For example, the sequence 2, 5, 8, 11 is an increasing arithmetic sequence because the terms increase by 3 as the index increases (5 – 2 = 3, 8 – 5 = 3, 11 – 8 = 3).

A decreasing arithmetic sequence is a sequence in which the terms of the sequence decrease as the index of the term increases. For example, the sequence 10, 7, 4, 1 is a decreasing arithmetic sequence because the terms decrease by 3 as the index increases (7 – 10 = -3, 4 – 7 = -3, 1 – 4 = -3).

It’s important to note that the common difference in an arithmetic sequence can be positive, negative, or zero, which determines whether the sequence is increasing, decreasing, or neither. For example, an arithmetic sequence with a common difference of -3 will be a decreasing sequence, while an arithmetic sequence with a common difference of 3 will be an increasing sequence. An arithmetic sequence with a common difference of 0 will be a constant sequence, where all terms are equal.

How to calculate the nth term and sum of the arithmetic sequence?

The nth term and sum of the arithmetic sequence can be evaluated with the help of formulas of the arithmetic sequence. Let us take some examples to learn how to calculate the problems of finding the nth term and sum of the sequence.

For finding the nth term

Example 1

Calculate the 23rd term of the arithmetic sequence of a given sequence.

1, 7, 13, 19, 25, 31, 37, 43, 49, ….

Solution

Step 1: Take the given sequence and find the common difference of the given sequence by taking the difference between two consecutive terms. 

1, 7, 13, 19, 25, 31, 37, 43, 49, ….

1st term = a₁ = 1

2nd term = a₂ = 7

Common difference = d = a₂ – a₁

Common difference = d = 7 – 1

Common difference = d = 6

The n value will be 23 as we have to evaluate the 23rd term of the sequence.

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

nthterm of the sequence= a_n = a₁ + (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 nth term.

23^rd term of the sequence = a₂₃ = 1 + (23 – 1) * 6

23^rd term of the sequence = a₂₃ = 1 + (22) * 6

23^rd term of the sequence = a₂₃ = 1 + 22 * 6

23^rd term of the sequence = a₂₃ = 1 + 132

23^rd term of the sequence = a₂₃ = 133

An nth-term calculator by Allmath can be used to find the sum or any term of the sequence according to the general formulas of an arithmetic sequence to avoid time-consuming calculations.

Example 2

Calculate the 19th term of the arithmetic sequence of a given sequence.

51, 46, 41, 36, 31, 26, 21, 16, 11, ….

Solution

Step 1: Take the given sequence and find the common difference of the given sequence by taking the difference between two consecutive terms. 

51, 46, 41, 36, 31, 26, 21, 16, 11, ….

1st term = a₁ = 51

2nd term = a₂ = 46

Common difference = d = a₂ – a₁

Common difference = d = 46 – 51

Common difference = d = -5

The n value will be 19 as we have to evaluate the 19th term of the sequence.

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

nthterm of the sequence= a_n = a₁ + (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 nth term.

19^th term of the sequence = a₁₉ = 51 + (19 – 1) * (-5)

19^th term of the sequence = a₁₉ = 51 + (18) * (-5)

19^th term of the sequence = a₁₉ = 51 + (18 * -5)

19^th term of the sequence = a₁₉ = 51 + (-90)

19^th term of the sequence = a₁₉ = 51 – 90

19^th term of the sequence = a₁₉ = -39

For finding the sum of the sequence

Example 1:

Find the sum of the first 15 terms of the given sequence.

12, 21, 30, 39, 48, 57, 66, 75, 84, …

Solution

Step 1: Take the given sequence and find the common difference of the given sequence by taking the difference between two consecutive terms. 

12, 21, 30, 39, 48, 57, 66, 75, 84, …

Initial term = a₁ = 12

Second term = a₂ = 21

Common difference = d = a₂ – a₁

Common difference = d = 21 – 12

Common difference = d = 9

The n will be 15 as we have to find the sum of the first 15 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 15 terms = s = 15/2 * (2(12) + (15 – 1) * 9)

Sum of the first 15 terms = s = 15/2 * (2(12) + (14) * 9)

Sum of the first 15 terms = s = 15/2 * (24 + 14 * 9)

Sum of the first 15 terms = s = 15/2 * (24 + 126)

Sum of the first 15 terms = s = 15/2 * 150

Sum of the first 15 terms = s = 15 * 75

Sum of the first 15 terms = s = 1125

Final words

Now you can grab all the basics of finding the nth term and the sum of the sequence from this post. We have covered the arithmetic sequence along with solved examples. Once you grab the basics of the arithmetic sequence you will master it.