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:
- Increasing arithmetic progression
- 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.
nth term of the sequence = xn = 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 = xn – xn-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 20th 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 = xn – xn-1
Let’s take the first two terms of the sequence.
x1 = 9
x2 = 14
Common difference = d = x2 – x2-1
= d = x2 – x1
= d = 14 – 9
= d = 5
Step III: Now write the general formula for the nth term of the arithmetic progression.
nth term of the sequence = xn = 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.
nth term of the sequence = xn = x1 + (n – 1) * d
put n = 20
20th term of the sequence = x20 = x1 + (20 – 1) * d
Now put x1 = 9 and d = 5
20th 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 187th 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 = xn – xn-1
Let’s take the first two terms of the sequence.
x1 = 13
x2 = 16
Common difference = d = x2 – x2-1
= d = x2 – x1
= d = 16 – 13
= d = 3
Step III: Now write the general formula for the nth term of the arithmetic progression.
nth term of the sequence = xn = 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.
nth term of the sequence = xn = x1 + (n – 1) * d
put n = 187
187th 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 = xn – xn-1
Let’s take the first two terms of the sequence.
x1 = 17
x2 = 22
Common difference = d = x2 – x2-1
= 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 = xn – xn-1
Let’s take the first two terms of the sequence.
x1 = 37
x2 = 47
Common difference = d = x2 – x2-1
= 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.
