What Is Arithmetic Progression? Formula, How To Calculate, Examples And Practices

In algebra, to find the correct sequence of the given data values or write a large number of terms in a single expression the arithmetic progression is frequently used. It makes a sequence of numbers by adding or subtracting similar terms to each successive number.

The sequence of the accurate order is determined by taking a common difference in the terms. The order of the sequence can be determined by using various methods. In this article, we will learn all the basics, calculations, and examples of the arithmetic progression just like we did on how to solve simultaneous equation using graphical method, substitution method, Elimination method, completing the square method and formula method.

This topic is a compulsory topic in mathematics examination and must be properly understood which is why the examples here have been well explained and properly broken down step by step.

What is the arithmetic progression?

A list of numbers or data values with a fixed pattern is said to be the arithmetic progression. The difference between two consecutive terms in the arithmetic progression is always constant.

If you take any number of data value from the list of numbers and subtract it from the previous number or data value, the difference must be similar each time.

In simple words, if the difference between any two consecutive terms of the sequence is constant then the sequence is said to be the arithmetic progression. For example, if the first term of the sequence is 12 and the common difference is 3 then the arithmetic progression that would be obtained must be:

12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, …

The constant or the similar term of the sequence is also known as the common difference. The well-known sets of numbers are the most commonly used examples of arithmetic progression. Such as the set of

• Natural numbers: The first term is 1 and the common difference is 1
• Whole numbers: The first term is 0 and the common difference is 1
• Even numbers: The first term is 2 and the common difference is 2
• Odd numbers: The first term is 2 and the common difference is 2
• Multiples of any natural number

The arithmetic progression can be:

1. Increasing arithmetic progression
2. Decreasing arithmetic progression

1. Increasing arithmetic progression

If the common difference between two consecutive terms of the list of numbers of a sequence is positive, then the sequence that would be obtained is said to be the Increasing arithmetic progression. For example, if the first term of the sequence is 5 and the common difference is 4

Then the obtained sequence must be:

5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, …

Hence, the above sequence is an increasing arithmetic progression.

2. Decreasing arithmetic progression

If the common difference between two consecutive terms of the list of numbers of a sequence is negative, then the sequence that would be obtained is said to be the decreasing arithmetic progression. For example, if the first term of the sequence is 25 and the common difference is -3

Then the obtained sequence must be:

25, 22, 19, 16, 13, 10, 7, 4, 1, -2, -5, -8, -11, -14, -17 …

Hence, the above sequence is a decreasing arithmetic progression.

Arithmetic Progression Formula

Let’s write the formulas to find the nth term of the sequence, the sum of the sequence, and the common difference.

i. The formula for finding the nth term

Below is the general formula for determining the nth term of the sequence.

n^th term of the sequence = x_n = x₁ + (n – 1) * d

ii. The formula for finding the sum of the sequence

Below is the general formula for evaluating the sum of the sequence

The sum of the sequence = S = n/2 * (2x₁ + (n – 1) * d)

iii. The formula for finding the common difference

Below is the general formula for calculating the constant or the common difference of the sequence.

Common difference = d = x_n – x_n-1

How to find the arithmetic progression?

The arithmetic progression can be determined easily by using its formulas. Let’s take a few examples of the arithmetic progression and learn how to calculate the arithmetic progression manually.

Examples for the nth term

Example 1:

Evaluate the 20^th term of the sequence by using the list of numbers 9, 14, 19, 24, 29, 34, 39, 44, 49, 54, …

Solution

Step I: Take the given list of numbers.

9, 14, 19, 24, 29, 34, 39, 44, 49, 54, …

The given data

nth term = 20

first term = x₁ = 9

Step II: Take the formula of the common difference and determine the constant term by taking the values from the given sequence of numbers.

Common difference = d = x_n – x_n-1

Let’s take the first two terms of the sequence.

x₁ = 9

x₂ = 14

Common difference = d = x₂ – x_2-1

= d = x₂ – x₁

= d = 14 – 9

= d = 5

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

n^th term of the sequence = x_n = x₁ + (n – 1) * d

Step IV: Substitute the given terms in the formula of the nth term to find the nth term of the sequence like the first term, constant term, and the nth term.

n^th term of the sequence = x_n = x₁ + (n – 1) * d

put n = 20

20^th term of the sequence = x₂₀ = x₁ + (20 – 1) * d

Now put x₁ = 9 and d = 5

20^th term of the sequence = x₂₀ = 9 + (20 – 1) * 5

= x₂₀ = 9 + 19 * 5

= x₂₀ = 9 + 95

= x₂₀ = 104

To ease up the calculations of determining the nth term, you can use an nth term calculator.

Example 2:

Evaluate the 187^th term of the sequence by using the list of numbers 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, …

Solution

Step I: Take the given list of numbers.

13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, …

The given data

nth term = 187

first term = x₁ = 13

Step II: Take the formula of the common difference and determine the constant term by taking the values from the given sequence of numbers.

Common difference = d = x_n – x_n-1

Let’s take the first two terms of the sequence.

x₁ = 13

x₂ = 16

Common difference = d = x₂ – x_2-1

= d = x₂ – x₁

= d = 16 – 13

= d = 3

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

n^th term of the sequence = x_n = x₁ + (n – 1) * d

Step IV: Substitute the given terms in the formula of the nth term to find the nth term of the sequence like the first term, constant term, and the nth term.

n^th term of the sequence = x_n = x₁ + (n – 1) * d

put n = 187

187^th term of the sequence = x₁₈₇ = x₁ + (187 – 1) * d

Now put x₁ = 13 and d = 3

= x₁₈₇ = 13 + (187 – 1) * 3

= x₁₈₇ = 13 + 186 * 3

= x₁₈₇ = 13 + 558

= x₁₈₇ = 571

Examples for sum of the arithmetic progression

Example 1

Evaluate the sum of the first 24 terms of the sequence by using the list of numbers 17, 22, 27, 32, 37, 42, 47, 52, 57, 62, 67, …

Solution

Step I: Take the given list of numbers.

17, 22, 27, 32, 37, 42, 47, 52, 57, 62, 67, …

The given data

nth term = 24

first term = x₁ = 17

Step II: Take the formula of the common difference and determine the constant term by taking the values from the given sequence of numbers.

Common difference = d = x_n – x_n-1

Let’s take the first two terms of the sequence.

x₁ = 17

x₂ = 22

Common difference = d = x₂ – x_2-1

= d = x₂ – x₁

= d = 22 – 17

= d = 5

Step III: Now write the general formula for the sum of the arithmetic progression.

The sum of the sequence = S = n/2 * (2x₁ + (n – 1) * d)

Step IV: Substitute the given terms in the formula to find the sum of the sequence like the first term, constant term, and the nth term.

Put n = 24

Sum of the sequence = S = 24/2 * (2x₁ + (24 – 1) * d)

Now put x₁ = 17 and d = 5

= 24/2 * (2(17) + (24 – 1) * 5)

= 24/2 * (34 + (24 – 1) * 5)

= 24/2 * (34 + 23 * 5)

= 24/2 * (34 + 115)

= 24/2 * 149

= 12 * 149

= 1788

Example 2

Evaluate the sum of the first 441 terms of the sequence by using the list of numbers 37, 47, 57, 67, 77, 87, 97, 107, 117, 127, 137, …

Solution

Step I: Take the given list of numbers.

37, 47, 57, 67, 77, 87, 97, 107, 117, 127, 137, …

The given data

nth term = 441

first term = x₁ = 37

Step II: Take the formula of the common difference and determine the constant term by taking the values from the given sequence of numbers.

Common difference = d = x_n – x_n-1

Let’s take the first two terms of the sequence.

x₁ = 37

x₂ = 47

Common difference = d = x₂ – x_2-1

= d = x₂ – x₁

= d = 47 – 37

= d = 10

Step III: Now write the general formula for the sum of the arithmetic progression.

The sum of the sequence = S = n/2 * (2x₁ + (n – 1) * d)

Step IV: Substitute the given terms in the formula to find the sum of the sequence like the first term, constant term, and the nth term.

Put n = 441

Sum of the sequence = S = 441/2 * (2x₁ + (441 – 1) * d)

Now put x₁ = 37 and d = 10

= 441/2 * (2(37) + (441 – 1) * 10)

= 441/2 * (74 + (441 – 1) * 10)

= 441/2 * (74 + 440 * 10)

= 441/2 * (74 + 4400)

= 441/2 * 4474

= 220.5 * 4474

= 986517

Summary

In this post, we have covered all the basics of the arithmetic progression along with a lot of solved examples. Now you can confirm that the arithmetic progression is not a difficult topic in mathematics.

You can solve any problem of the sum of the sequence or nth term of the sequence easily just by learning the above post.