CBSE Class 10 Maths Revision Notes Chapter 5

Class 10 Mathematics Revision Notes for Arithmetic Progressions of Chapter 5

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Class 10 Mathematics Revision Notes for Arithmetic Progressions of Chapter 5

Access Class 10 Mathematics Chapter 5 – Arithmetic Progression Notes

Definition of Arithmetic Progression:

• A sequence of numbers which we get by adding a fixed number to the preceding term starting from the first term such that the difference between each consecutive term remains the same.
• The fixed term is called the common difference of the arithmetic progression. 
• The common difference can be positive, zero or negative.
• Each number in the list of arithmetic progressions is called a term. 

General term of an AP:

The general form of an arithmetic progression is: 

a, a + d, a + 2d, a + 3d, ……….., a + (n -1)d

A.P. can be of two types. It is as follows: 

Finite AP: A finite A.P. is one which has a finite number of terms with a + (n -1)d as the last term.

Example: 1, 3, 5, 7, ………,25

Infinite AP: An infinite A.P. is one which neither has a finite number of terms nor has the last term.

Example: 2,4,6,8……...∞

Sum of the terms of an AP:

The sum of n terms of an A.P. if the first term and common difference is given: 

S = n/2 (2a + (n -1) d)

The sum of n terms of an A.P. if the first term and last term l is given: 

S = n/2 (a + l)

Arithmetic Progression 

Common Difference

The difference between the two consecutive terms of an A.P. is known as the common difference. 

For example: 3,6,9,12…….. The common difference here is 3. 

Classification of the common differences:

• The common difference can be negative when the A.P. is decreasing. 
• It can be positive when the A.P. is increasing. 
• The common difference will be zero when the A.P. is constant. 

General Form of an Arithmetic Progression 

The general form of an A.P. is (a, a + d, a + 2d, a + 3d……..) where a is considered the first term and d is the common difference. In the given condition, d = 0 OR d > 0 or d < 0. 

Finite and Infinite A.P.

Finite AP: An A.P. in which the number of terms is finite. Example: 2, 5, 8…….32, 38.

Infinite AP: An A.P. in which there are an infinite number of terms. Example: 2, 4, 6, 7, 9……

Sum of Arithmetic Progressions

The sum of n terms of an A.P. with a as its first term and d as a common difference is written as: 

Sn = n/2 (2a + (n-1)d)

Arithmetic Mean

The arithmetic mean is the average of a given set of numbers. It is calculated as: 

AM = Sum of terms/Number of terms 

For example: if x, y and z are in arithmetic progression, then their mean will be y = (x + z)/2. Here, y is the arithmetic mean of x and z. 

Properties of Arithmetic Progressions

• If the same number is added or subtracted from each A.P. term, the resulting terms are also in A.P. with the identical common difference. 
• If each term in A.P. is divided or multiplied by the same non-zero number, then the resulting series will also be in A.P. 
• The three numbers x, y and z  will be an A.P. when 2y = x + z. 
• If the nth term of a series is a linear expression, then the series is an A.P.
• If we choose terms from the A.P. in a regular interval, the chosen terms will also constitute an A.P.
• If the terms of an A.P. are increased or decreased with the same amount,  the resulting sequence will also be an A.P.

Solved Examples

Example 1: 

For the AP: 3/2, 1/2, -½, -3/2, …..  write the first term a and the common difference d.

Solution: 

Here a = 3/2, d = 1

Example 2: 

Find the 10th term of the AP: 2, 7, 12, …..

Solution: 

Here, a = 2, d = 7-2 = 5 and n = 10.

We have an = a + (n – 1)d

So a10 = 2 + (10 – 1) X 5 = 2 + 45 = 47 

Example 3: 

Find the sum of the first 22 terms of the AP: 8, 3, -2, …..

Here, a = 8, d = 3 – 8 = –5, and n = 22.

Since, S = n/2 [2a + (n – 1)d]

Therefore, S = 22/2 [16 + 21 (-5)] = 11 (16 – 105) = 11(-89) = -979 

Hence the sum of the first 22 terms of the A.P. will be – 979.

Important Points on Arithmetic Progression

• If each term in the A.P. is increased, decreased, multiplied, or divided by the same non-zero constant, then the following sequence will be in A.P.
• In the A.P., all the terms that are equidistant from the start to end will be constant. 
• To solve most of the problems related to A.P., the terms can be taken as:

3 terms: (a – d), a, (a + d)

4 terms: (a – 3d), (a – d), (a + d), (a + 3d)

Important formulas in AP