The term arithmetic progression is widely in set theory and algebra. The set theory is a mathematical theory of well-determined collections of objects that are known as members of the set.
The arithmetic sequence finds the sequence of the given terms. The sequence will be consistent as the difference among the members is constant.
In this article, we are going to explore the basics of the arithmetic progression along with methods and examples.
Definition of Arithmetic Progression.
In mathematics, the arithmetic progression is the ordered pair of the set of observations that have constant differences among each consecutive term.
Two successive terms remain the same in arithmetic sequence. By taking the first term and the common difference between the terms, the arithmetic sequence can be determined easily. To make the sequence simply add the constant term to the previous term of the sequence.
The common difference is another name for the constant term. The even numbers, odd numbers, natural numbers, whole numbers, and integers, are common examples of the arithmetic sequence as the common difference is the same throughout the sequence.
Kinds of the Arithmetic Progression
Here are the two kinds of the arithmetic sequence:
- Increasing Arithmetic Progression
- Decreasing Arithmetic Progression
The kinds of the sequence are dependent on the value of the common difference. Here is a brief introduction to the types of an arithmetic sequences.
1. Increasing Arithmetic Progression
The increasing arithmetic progression is that sequence in which the constant term is positive.
The positive constant term causes an increase in the sequence as it goes from least to greatest that is why it is known as the increasing arithmetic progression.
If the first term of the sequence is 3 and the common difference is 10 then the sequence should be:
3, 13, 23, 33, 43, 53, 63, 73, 83, 93, 103, ….
The sequence is increasing as the value goes from least to greatest.
2. Decreasing Arithmetic Progression
The decreasing arithmetic progression is that sequence in which the constant term is negative. The negative constant term causes a decrease in the sequence as it goes from greatest to least that is why it is known as the decreasing arithmetic progression.
If the first term of the sequence is 33 and the common difference is -3 then the sequence should be:
33, 30, 27, 24, 21, 18, 15, 12, 9, 6, 3, ….
The sequence is decreasing as the value goes from greatest to least.
Formulas of Arithmetic Progression
There are different formulas for the arithmetic progression for finding:
- For nth term
- For the sum of the sequence
- For finding the common difference
Let’s have a look at these formulas.
I. For the nth term
The formula of the arithmetic progression for finding the nth term of the sequence is:
nthterm of the sequence= pn = p1 + (n – 1) * d
- pn = the nth term
- p1 = the first term of the sequence
- n = total number of terms
- d = common difference
II. For the sum of the sequence
The formula of the arithmetic progression for finding the sum of the sequence is:
Sum of the sequence = s = n/2 * (2p1 + (n – 1) * d)
- p1 = the first term of the sequence
- n = total number of terms
- d = common difference
III. For finding the common difference
The formula of the arithmetic progression for finding the common difference.
Common difference = d = pn – pn-1
How to calculate the arithmetic progression?
The arithmetic progression can be determined easily either by using a manual method or an online calculator. Let us discuss this briefly.
· By using Calculator
There are several online tools are available for solving all kinds of numerical problems.
An nth term calculator can be used to solve the problems of the arithmetic progression with steps. Follow the below steps to use this calculator.
- Enter the required values into the required input fields.
- Click the calculate button
- The result of the nth term and the sum of the sequence will come in a couple of seconds.
· Manually
Example 1: For the nth term
Evaluate the 21th term of the sequence, if the arithmetic progression is 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, …
Solution
Step-I: Take the given arithmetic progression.
17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, …
Step-II: Calculate the common difference of the sequence and write the first term & nth term.
n = 21
p1 = 17
p2 = 21
Common difference = d = p2 – p1
= 21 – 17
= 4
Step-III: Take the nth term formula of the arithmetic progression and substitute the common difference, first term, and the nth term in the sequence.
nthterm of the sequence= pn = p1 + (n – 1) * d
21th term of the sequence= a21 = p1 + (21 – 1) * d
= 17 + (21 – 1) * 4
= 17 + (20) * 4
= 17 + 80
= 97
Example 2
Evaluate the 111th term of the sequence, if the arithmetic progression is 18, 22, 25, 30, 34, 38, 42, 46, 50, 54, 58, 62, …
Solution
Step-I: Take the given arithmetic progression.
18, 22, 25, 30, 34, 38, 42, 46, 50, 54, 58, 62, …
Step-II: Calculate the common difference of the sequence and write the first term & nth term.
n = 111
p1 = 18
p2 = 22
Common difference = d = p2 – p1
= 22 – 18
= 4
Step-III: Take the nth term formula of the arithmetic progression and substitute the common difference, first term, and the nth term in the sequence.
nthterm of the sequence= pn = p1 + (n – 1) * d
111th term of the sequence= a111 = p1 + (111 – 1) * d
= 19 + (111 – 1) * 4
= 19 + (110) * 4
= 19 + 440
= 459
Example 3: For the sum of the sequence
Evaluate the sum of the first 25 terms of the sequence, if the arithmetic sequence is 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, …
Solution
Step-I: Take the given arithmetic progression.
5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, …
Step-II: Calculate the common difference of the sequence and write the first term & nth term.
n = 25
p1 = 5
p2 = 11
Common difference = d = p2 – p1
Common difference = d = 11 – 5
Common difference = d = 6
Step-III: Take the sum of the sequence formula of the arithmetic progression and substitute the common difference, first term, and the nth term in the sequence.
Sum of the sequence = s = n/2 * (2p1 + (n – 1) * d)
= 25/2 * (2p1 + (25 – 1) * d)
= 25/2 * (2(5) + (25 – 1) * 6)
= 25/2 * (2(5) + (24) * 6)
= 25/2 * (10 + (24) * 6)
= 25/2 * (10 + 144)
= 25/2 * 154
= 12.5 * 154
= 1925
Final Words
Now you can grab all the basics of the arithmetic progression from this article. We have covered all the basics like definition, kinds, formulas, and solved examples in this post.