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Arithmetic progression is a sequence of numbers where the numbers follow an order and have a constant difference between them. An arithmetic progression is also known as an arithmetic sequence, where the difference between the previous and next term is the same. In an arithmetic progression, the next term is generally determined by applying some arithmetic operations. An arithmetic progression is required in many fields, like physics, engineering, economics, etc., for predicting sequences or analysing trends by observing the rate of change.
These notes covered all the topics and subtopics of arithmetic progression, like the basic definition of arithmetic progression, finite and infinite arithmetic progression, general term of arithmetic progression, sum of the nth term and arithmetic mean and formulae required to calculate the arithmetic progression. CBSE Class 10 chapter Arithmetic Progression also includes the nth term of the arithmetic progression. Students must practice all the topics of arithmetic progression and their examples from the NCERT Exemplar Solutions for Class 10 Maths Chapter 5 Arithmetic Progression. Students must practice questions and check their solutions using the NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progression.
Arithmetic Progression: Arithmetic progression is a sequence, list or series of numbers where the difference between two terms is constant, and we can get the preceding and succeeding terms by performing the arithmetic operation.
Arithmetic progression is denoted by
Example: 5, 8, 11, 14, 17, 20, 23......
Common difference: In an arithmetic progression, the difference between two consecutive terms is called the common difference, and it is denoted by
Example: 7, 12, 17, 22, 27, 32
In this arithmetic progression, the common difference (
Formula For Calculating Common Difference:
Where,
There are three types of common difference
1. Positive Difference: The
Example: 1, 2, 3, 4, 5
2. Negative Difference: In this case, the
Example: 100, 90, 80, 70, 60, 50
3. Zero Difference: In this case, the
Example: 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3
There are two types of arithmetic progression (AP):
1. Finite Arithmetic Progression: An arithmetic progression (AP) in which the number of terms is finite is called a finite arithmetic progression (AP). The finite AP have an nth term.
Example: 1, 3, 5, 7, 9, 11
2. Infinite Arithmetic Progression: An arithmetic progression (AP) in which the number of terms is infinite is called an infinite arithmetic progression (AP). The infinite AP does not have the nth term.
Example: 2, 4, 6, 8, 10, ........
The nth term in the last term in AP and the formula for determining the nth term is as follows:
Here,
a = First term
d = Common difference
n = number of terms
Example: Determine the 10th term of the arithmetic progression 4, 8, 12, 16, .......
Given:
AP = 4, 8, 12, 16, .......
Here, a = 4
(8 - 4) = 4 and (12 - 8) = 4
So, d = 4
And now, we have to find the 10th term of the given AP.
Therefore, n = 10
After substituting the values in the formula, we get
a10 = 4 + (10 – 1)4
a10 = 4 + 9 × 4
a10 = 4 + 36
a10 = 40
So, the 10th term of the given arithmetic sequence 4, 8, 12, 16, ....... is 40
The general form of an AP is as follows:
a, a + d, a + 2d, a + 3d, a + 4d ....
Where a is the first term and d is the common difference, the value of d = 0, d > 0 or d < 0.
The formula for calculating the sum of the terms in AP is as follows:
Where,
S = Sum of the terms
n = Number of terms
a = First term
d = Common difference
Or we can also write this as,
Example: Determine the sum of the first 18 terms of AP 17, 11, 5, -1, .......
Given:
AP = 17, 11, 5, -1, .......
First term a = 17
Common difference d = 11 - 17 = -6, and 5 - 11 = -6
Therefore, the common difference d = -6
Number of terms n = 18
After substituting all the values, we get
Therefore, the sum of the 18th term is -612.
The average of the terms in an arithmetic progression is called the Arithmetic Mean (AM). Therefore, the arithmetic mean formula is as follows:
The sum of the first n natural numbers can be determined by:
Here,
n = Number of terms
Example: Determine the sum of the first 15 natural numbers.
Given:
n = 15
Therefore, the sum of the first 15 natural numbers is 120.
Students must download the notes below for each chapter to ace the topics.
NCERT Class 10 Solutions for Maths And Science
Students must check the NCERT solutions for Class 10 Maths and Science given below:
Students must check the NCERT exemplar solutions for Class 10 Maths and Science given below:
Read More: Syllabus and Books
An arithmetic progression is a sequence, list or series of numbers where the difference between two terms is constant, and we can get the preceding and succeeding terms by performing the arithmetic operation.
The important formulas in Arithmetic progression are:
The common difference in an Arithmetic Progression (AP) =
The nth term of an Arithmetic Progression (AP) =
The sum of the terms in Arithmetic Progression (AP) =
Arithmetic Mean =
The sum of the n natural numbers =
Students can determine that the nth term of the arithmetic progression is
The formula for calculating the sum of the terms in AP is
In an arithmetic progression, the difference between two consecutive terms is called the common difference, and it is denoted by
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