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# Arithmetic Progressions Class 10 Notes CBSE Maths Chapter 5 (Free PDF Download)

Last updated date: 11th Aug 2024
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## Class 10 Maths Revision Notes for Arithmetic Progressions of Chapter 5 - Free PDF Download

We at Vedantu have prepared the notes for Arithmetic Progression Class 10 to help students revise the whole topic effectively. The highly experienced teachers at Vedantu have prepared the CBSE Solutions and other study materials on Arithmetic Progression. These study materials are available in PDF format and students can download the PDF files for free. You can download these study materials from our Vedantu app as well. The Arithmetic Progression Class 10 Maths Revision Notes for Chapter 5 will help you to revise the whole topic thoroughly. Hence you can understand the concept and secure good marks in the exam. Subjects like Science, Maths, English will become easy to study if you have access to NCERT Solution Class 10 Science, Maths solutions, and solutions of other subjects that are available on Vedantu only.

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## Access Class 10 Maths Chapter 5 – Arithmetic Progression Notes

### Definition of Arithmetic Progression:

• An arithmetic progression is a sequence of numbers, obtained 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.

• Each of the numbers in the list is called a term and the fixed number is called the common difference of the AP which can be any integer.

• For example: $2,5,8,11....$ having a common difference of $3$.

### General term of an AP:

1. The general form of an AP is:

$a\text{ },a+d\text{ },a+2d\text{ },a+3d\text{ },....,a+(n-1)d$

2. An AP with a finite number of terms is called a finite AP having           $a+(n-1)d$ as the last term.

For example:

Finite AP: $1,3,5,7,....,25$

An AP which neither has a finite number of terms nor has a last term is called an infinite AP.

For example:

Infinite AP: $2,4,6,8.....\infty$

3. The ${{n}^{th}}$ term of the AP: $an=a+(n-1)d$, where $a$ is the first term of the sequence and $d$ is the common difference.

The Second term: ${{a}_{2}}=a+(2-1)d=a+d$

Similarly, the third term ${{a}_{3}}=a+(3-1)d=a+2d$

The fourth term ${{a}_{4}}=a+(4-1)d=a+3d$ and so on till the last term.

Example 1:

An AP has a first term $3$, common difference $4$. Find the third and fifth term of the AP.

Ans:

$a=3,\text{ d}=\text{4}$

${{\text{a}}_{3}}=3+(3-1)4$

${{a}_{3}}=11$

Similarly,

${{a}_{5}}=3+(5-1)4$

${{a}_{5}}=19$

4. ${{n}^{th}}$ term of an AP from the end:  $tn=L-(n-1)d$, where $\text{L}$ is the last term of the AP.

Example 2:

An AP has a common difference $2$ and last term $24$. Find the fourth term of the AP from the end.

Ans:

$d=2,\text{ L}=2\text{4}$

${{t}_{4}}=24-(4-1)2$

${{t}_{4}}=18$

### Sum of the terms of an AP:

1. Sum of $n$ terms of an AP if first term and common difference is given:

$S=\dfrac{n}{2}(2a+(n-1)d)$

1. Sum of $n$ terms of an AP if first term and last term $l$ is given:

$S=\dfrac{n}{2}(a+l)$

Example 3:

Find the sum of first $10$ terms of the AP $1,4,7,10.....34.$

Ans:

$S=\dfrac{10}{2}(2\times 1+(10-1)3)$

$=5(2+27)$

$=5\times 29$

$=145$

## Arithmetic Progression

An arithmetic progression is a sequence of numbers that differ from each other by a common difference. For example, the sequence 3, 6, 9, 12, ..... is an A.P. with a common difference of 3.

### Common Difference

The difference between the two consecutive terms of an A.P. is known as the common difference. For example, in the sequence 3, 6, 9, 12...., the common difference is 3.

The classification of the common difference:

• Positive, when the A.P. is increasing.

• Zero, when the A.P. is constant.

• Negative, when the A.P. is decreasing.

### General Form of an Arithmetic Progression

Say the terms a1, a2, a3……an are in A.P. If the first term is ‘a’ and its common difference is ‘d’. Then, the terms can also be expressed as follows.

1st term a1 = a

2nd term a2 = a + d

3rd term a3 = a + 2d

Therefore, we can also represent arithmetic progressions as:

a, a + d, a + 2d, ……

This representation is called the general form of an Arithmetic Progression.

### Finite and Infinite A.P.

• In the finite A.P., the numbers of terms are finite, and the last term of the A.P. exists.

• In the infinite A.P., the number of terms is infinite, and the last term of an A.P. doesn’t exist.

### Sum of Arithmetic Progressions

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

$S_{n} = \frac{n}{2} (2a + (n - 1)d)$

### Arithmetic Mean

Arithmetic Mean is simply the average of two numbers. If we have two numbers n and m, we can add a number L in between them so that the three numbers form an arithmetic sequence like n, L, m. In this case, the number L is the arithmetic mean of the numbers n and m. On the basis of the properties of Arithmetic Progression, we may say:

L – n = m – L, that is, the arithmetic mean of n and m.

$L = \frac{n + m}{2}$

### Properties of Arithmetic Progressions

• If the same number is added or subtracted from each A.P. term, the resulting terms in the sequence are also in A.P. with the same common difference.

• If each term in A.P. is divided or multiplied by the same non-zero number, the resulting series is also in A.P.

• Three numbers x, y, and z will be an A.P. if 2y = x + z.

• A series is an A.P. if the nth term is a linear expression.

• If we pick terms from the A.P. in the regular interval, these selected terms will also make an A.P.

• If the terms of an arithmetic progression are increased or decreased with the same amount, the resulting sequence will also be an arithmetic progression.

### Solved Examples

1. Find the missing term in the following AP. 5, x, 13.

Ans: Three numbers x, y, z are in AP if 2y = x + z. Given x = 5, z = 13 therefore y = (x+z)/2.

By substituting values we get:

y = (5 + 13)/2 = 18/2 = 9

Therefore, the missing term is 9.

2. Write the first five terms of A.P., when a = 23 and d = - 5.

Ans: The general form of A.P. is a, a+d, a+2d …

When a = 23 and d = -5, the first five terms of AP are 23, 23+(-5), 23 + 2(-5), 23 +3(-5), 23 + 4(-5).

Hence, the first five terms of AP are 23, 18, 13, 8 and 3.

### Important Points on Arithmetic Progressions

• If each term of the A.P. is increased, decreased, multiplied, or divided by the same non-zero constant, the resulting sequence would also be in A.P.

• In the A.P., the number of terms equidistant from start to end will be constant.

• In order to solve most of the problems related to A.P., the terms can be conveniently taken as:

3 Terms: (a - d), a, (a + d).

4 Terms: (a - 3d), (a - d), (a + d), (a + 3d).

### Important Formulas

 First Term of AP a, a + d, a + 2d, a + 3d, a + 4d, ………. ,a + (n – 1) d Common Difference in Arithmetic Progression d = a2 – a1 = a3 – a2 = ……. = an – an – 1 nth Term of an AP an = a + (n − 1) × d Sum of N Terms of AP $S_{n} = \frac{n}{2} (2a + (n - 1)d)$

### CBSE Class 10 Revision Notes - Other Chapters

The following is a list of links to the revision notes for all the chapters included in the CBSE Class 10 syllabus. We advise students to visit the mentioned pages to reap the benefits of Vedantu’s expert-curated materials.

## What are the Benefits of Referring to Vedantu’s Revision Notes for Class 10 Maths Chapter 5 - Arithmetic Progression

• Provides quick, clear summaries of key concepts.

• Simplifies complex topics for better understanding.

• Efficient tool for last-minute exam prep.

• Enhances retention of crucial information.

• Supports effective exam preparation with key points and tips.

• Saves time by consolidating information.

• Prioritizes important topics and questions.

• Offers practical examples for real-world connections.

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## Conclusion

Arithmetic Progressions holds immense importance in CBSE Class 10 Maths. Vedantu's expert teachers have curated comprehensive study materials and essential notes, offering students thorough preparation for the Class 10 exam. These revision notes cover all vital information about the chapter, ensuring students are well-equipped to score high marks. We strongly recommend students to delve into these notes and explore related links in this article to derive the best possible results in their preparation for CBSE Class 10 Maths Chapter 5 - Arithmetic Progressions.

## FAQs on Arithmetic Progressions Class 10 Notes CBSE Maths Chapter 5 (Free PDF Download)

1. Find the Sum of the First Ten Numbers of this Arithmetic Series: 1, 11, 21, 31?

We can use this formula:

S = 1/2 (2a1 + d(n-1))n

Given a1 = 1, d = 10, n = 10

S = 1/2 (2.1 + 10(10-1))10 = 5(2 + 90) = 5.92 = 460

2. The First Three Terms of A.P. are (3y − 1). (3y + 5) and (5y + 1). Then y Equals to?

Given that, The first three terms of A.P. are (3y − 1), (3y + 5), (5y + 1).

Therefore,

2(3y + 5) = 3y − 1 + 5y + 1

⇒ 6y + 10 =8y

⇒ 2y = 10

⇒ y = 5

Hence, the value of y is 5.

3. What is an arithmetic progression according to Revision Notes of Chapter 5 of Class 10 Maths?

An arithmetic progression is a list of numbers in which each individual term of the progression can be obtained by adding a fixed number to the preceding term. This can be followed for all terms except the first term. This fixed number that is added to the aforementioned terms is known as the common difference of the Arithmetic Progression (AP). The common difference can be either positive, negative or zero.

4. What are finite and infinite APs according to Revision Notes of Chapter 5 of Class 10 Maths?

In some arithmetic progressions, there are only a finite number of terms. The last term is part of the sequence with the help of which the total number of terms can be counted. This is called a finite AP. In contrast to this, some APs do not have a last term and the total number of terms cannot be counted either. These are referred to as infinite APs. For more information on different types of APs, you should check out Revision Notes of Chapter 5 of Class 10 Maths by Vedantu. They are free of cost and also available on Vedantu mobile app.

5. How many exercises in total are present in Chapter 5 of Class 10 Maths?

In chapter 5 of the NCERT Class 10 Maths textbook, four exercises are present in total. The first exercise asks you to identify whether the sequence provided is an AP and extract the first term a and the common difference d. Exercise 5.2 deals with more questions asking you about the nth term of the AP. Exercise 5.3 is all about the summation of the first n terms of an AP while exercise 5.4 is optional.

6. Where can I get Revision Notes of Chapter 5 of Class 10 Maths?

Vedantu is known to have the best repository of all study material that can ever be required by students of any class. They provide quality content curated by expert teachers across the nation with years of teaching experience. For Class 10 Maths too, you can rely on Vedantu. Download Revision Notes of Class 10 Maths for Arithmetic Progressions free of cost.

7. Why is Chapter 5 of Class 10 Maths important?

Studying Arithmetic Progression helps you analyse certain patterns that present themselves in our daily life. An AP is a sequential pattern of numbers in which the difference between consecutive terms is constant. This chapter lays the foundation for several chapters taught in higher classes too. Although only basics are taught in Class 10, it is important to understand each concept taught to you in order to do well in exams too.