How to calculate arithmetic sequence? Explained with examples

The arithmetic sequence is a well-known method used to evaluate the sequence and sum of the sequence of the terms. This technique is frequently used in various branches of mathematics. The letters & symbols are used in this technique to represent numbers in the form of an equation. 

The collection of things or objects that are well determined is also referred to as arithmetic sequence that is used in set theory for various purposes. In this lesson, we will go through the basics of the arithmetic sequence along with the definition, expression, and examples. 

What is the arithmetic sequence?

The arithmetic sequence is a wide concept that is used in algebra and set theory and is defined as a sequence of numbers or integers in which the difference among each consecutive number is the same. The difference between the numbers is known as the common difference. 

The common difference is the difference between two consecutive terms in a sequence that gives always the same result. The sequence can be created or the given sequence can be summed up with the help of the expression of the arithmetic sequence. 

Such as, 

Sequence of numbers = 12, 25, 38, 51, 64, 77, 90, … 

The above sequence of numbers is an arithmetic sequence as the difference between each consecutive number is the same that is 13. You can also create a sequence of the numbers if the first term and the common difference are given. Such as,

If the first term is 12 and the common difference is 3 then 

12, 15, 18, 21, 24, 27, 30, …

Order of sequence

There are two forms of the order of sequence.

  • Increasing sequence 
  • Decreasing sequence

The order of the arithmetic sequence is depending on the nature of the constant difference of the given sequence. Let us briefly describe the orders of the arithmetic sequence.

  1. Increasing sequence
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The sequence should be increasing if the values or numbers go from least to greatest such as forming an ascending order. If the constant difference is positive, then it always makes an increasing sequence as the values should go from the least to the greatest.  

For example, if the initial number of the sequence is 12 and the common difference is 7 then the sequence that is obtained is:

19, 26, 33, 40, 47, 54, 61, 68, 75, …

The numbers of the sequence make an ascending order as the constant difference is positive then we can say that the given sequence is increasing.

  1. Decreasing sequence

The sequence should be decreasing if the values or numbers go from greatest to least such as forms a descending order. If the constant difference is negative, then it always makes a decreasing sequence as the values should go from the greatest to the least.  

For example, if the initial number of the sequence is 29 and the common difference is -7 then the sequence that is obtained is:

29, 22, 15, 8, 1, -6, -13, -20, …

The numbers of the sequence make a descending order as the constant difference is negative then we can say that the given sequence is decreasing.

Expression of an arithmetic sequence

There are three different expressions that are used in arithmetic sequences for the calculation of various terms. These terms are

  1. For nth term 
  2. For the sum of the sequence 
  3. For finding the common difference

The expressions for calculating the above terms are very useful and helpful for the calculation of the problems of the arithmetic sequence manually.

An arithmetic sequence calculator can be used to solve the problems of arithmetic sequence according to the formulas to avoid lengthy calculations. 

  1. Expression for finding the nth term of the sequence
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The nth term of the sequence has a general expression such as:

nth term of the sequence= pn = p1 + (n – 1) * d

where 

  • pn is the nth term of the sequence 
  • p1 is the first term of the sequence 
  • n is the total number of terms 
  • d is the constant difference. 
  1. Expression for finding the sum of the sequence

The sum of the sequence has a general expression such as:

Sum of the sequence = s = n/2 * (2p1 + (n – 1) * d)

Where 

  • f1 is the starting term of the sequence 
  • n is the total number of terms 
  • d is a common difference. 
  1. Expression for finding the common difference 

Here is the general expression for calculating the common difference of the sequence. 

Common difference = d = pn – pn-1 

How to calculate the arithmetic sequence?

The expressions of the arithmetic sequence are used to calculate the problems of the arithmetic sequence. Let us take a few examples to learn how to calculate the arithmetic sequence. 

Example 1: For finding the nth term

Evaluate the 15th term of the sequence by taking the values from the given sequence,

2, 9, 16, 23, 30, 37, 44, 51, …

Solution 

Step 1: First of all, evaluate the common difference and take out the initial term from the given sequence of numbers.  

2, 9, 16, 23, 30, 37, 44, 51, …

Initial term = p1 = 2

Second term = p2 = 9

Common difference = d = p2 – p

Common difference = d = 9 – 2

Common difference = d = 7

We have to calculate the 25th term of the sequence so n = 25

Step 2: Now take the general expression for finding the nth term of the sequence.   

nth term of the sequence= pn = p1 + (n – 1) * d

Step 3: Put the values of the constant difference and the first term into the sequence to calculate the nth term of the sequence.

15th term of the sequence= a15 = 2 + (15 – 1) * 7

15th term of the sequence= a15 = 2 + (14) * 7

15th term of the sequence= a15 = 2 + 98

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15th term of the sequence= a15 = 100

Example 2: For finding the sum of the sequence

Evaluate the sum of the first 9 terms of the sequence by taking the values from the given sequence,

7, 16, 25, 34, 43, 52, 61, 70, 79, …

Solution 

Step 1: First of all, evaluate the common difference and take out the initial term from the given sequence of numbers.  

7, 16, 25, 34, 43, 52, 61, 70, 79, …

Initial term = p1 = 7

Second term = p2 = 16

Common difference = d = p2 – p

Common difference = d = 16 – 7

Common difference = d = 9

We have to calculate the sum of the first 30 terms of the sequence so n = 30

Step 2: Now take the general expression for finding the sum of the sequence.   

Sum of the sequence = s = n/2 * (2p1 + (n – 1) * d)

Step 3: Put the values of the constant difference and the first term into the sequence to calculate the sum of the sequence.

Sum of the first 9 terms = s = 9/2 * (2(7) + (9 – 1) * 9)

Sum of the first 9 terms = s = 9/2 * (14 + (9 – 1) * 9)

Sum of the first 9 terms = s = 9/2 * (14 + 8 * 9)

Sum of the first 9 terms = s = 9/2 * (14 + 72)

Sum of the first 9 terms = s = 9/2 * 86

Sum of the first 9 terms = s = 4.5 * 86

Sum of the first 9 terms = s = 387

Final words

Now you can solve any problem for finding the nth term and sum of the sequence easily just by following the above post. We have discussed each and every basic of finding the arithmetic sequence along with solved examples.

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