CBSE Class 10 Maths Chapter 20: Arithmetic Progressions NCERT Solutions

Reviewed by · Updated on

NCERT Solutions PDF Class 10 PDF

This comprehensive set of NCERT Solutions for CBSE Class 10 Mathematics, Chapter 20, focuses on Arithmetic Progressions (AP). It provides detailed explanations and step-by-step solutions for the exercises, helping students grasp the fundamental concepts of AP. The chapter covers identifying APs, understanding the first term (a) and common difference (d), and calculating subsequent terms. The solutions clarify the conditions under which a sequence of numbers forms an AP, emphasizing the constant difference between consecutive terms. This resource is designed to aid students in understanding the core principles of arithmetic progressions and preparing effectively for their examinations by offering clear, concise, and accurate problem-solving guidance.

Home

Quick info

BoardCBSE
ClassClass 10
SubjectMathematics
Session2026
LanguageEnglish
TypeNCERT Solutions
ChapterChapter 20

Chapter summary

Chapter 20 of the CBSE Class 10 Mathematics syllabus introduces Arithmetic Progressions (AP). These NCERT Solutions cover the initial concepts, including how to determine if a given sequence of numbers forms an AP by checking for a common difference. The solutions also guide students on constructing the first few terms of an AP when the first term and common difference are provided. This chapter lays the groundwork for more advanced AP concepts in subsequent sections.

Learning outcomes

  • Understand the definition and properties of an Arithmetic Progression (AP).
  • Identify whether a given sequence of numbers forms an AP.
  • Determine the common difference of an AP.
  • Calculate the first four terms of an AP given the first term and common difference.
  • Differentiate between situations that form an AP and those that do not.

Topics covered

Paper topics

  • Arithmetic Progression (AP)
  • Identifying APs
  • Common Difference
  • First Term
  • Calculating Terms of AP
  • Real-world applications of AP

Important topics

  • Definition of Arithmetic Progression
  • Condition for forming an AP
  • Calculating terms using first term and common difference
  • Distinguishing AP from non-AP sequences

PDF preview

Read page by page below. PDF is streamed from the official NCERT website — no download button on this page.

Loading document …
Page of
Loading page …

Questions and Solutions

Question 1

In which of the following situations, does the list of numbers involved make an arithmetic progression and why?

(i) The taxi fare after each km when the fare is Rs 15 for the first km and Rs 8 for each additional km.

(ii) The amount of air present in a cylinder when a vacuum pump removes ¼ of the air remaining in the cylinder at a time.

(iii) The cost of digging a well after every metre of digging, when it costs Rs 150 for the first metre and rises by Rs 50 for each subsequent metre.

(iv) The amount of money in the account every year, when Rs 10000 is deposited at compound interest at 8% per annum.

Solution:

To determine if a list of numbers forms an Arithmetic Progression (AP), we need to check if the difference between consecutive terms is constant. This constant difference is known as the common difference (d).

(i) Taxi Fare:

  • Fare for the 1st km = Rs 15
  • Fare for the first 2 km = Rs 15 + Rs 8 = Rs 23
  • Fare for the first 3 km = Rs 23 + Rs 8 = Rs 31
  • Fare for the first 4 km = Rs 31 + Rs 8 = Rs 39

The sequence of fares is 15, 23, 31, 39, ...

The difference between consecutive terms is: 23 - 15 = 8, 31 - 23 = 8, 39 - 31 = 8. Since the difference is constant (d=8), this situation forms an AP.

(ii) Air in a Cylinder:

Let the initial volume of air be V.

  • After 1st removal: V - (1/4)V = (3/4)V
  • After 2nd removal: (3/4)V - (1/4)(3/4)V = (3/4)V * (1 - 1/4) = (3/4)V * (3/4) = (3/4)^2 V
  • After 3rd removal: (3/4)^2 V - (1/4)(3/4)^2 V = (3/4)^2 V * (3/4) = (3/4)^3 V

The sequence of air volumes is V, (3/4)V, (3/4)^2 V, (3/4)^3 V, ...

The difference between consecutive terms is: (3/4)V - V = -(1/4)V, and (3/4)^2 V - (3/4)V = (3/4)V * (3/4 - 1) = (3/4)V * (-1/4) = -(3/16)V. Since the differences are not constant, this situation does not form an AP.

(iii) Cost of Digging a Well:

  • Cost for the 1st metre = Rs 150
  • Cost for the first 2 metres = Rs 150 + Rs 50 = Rs 200
  • Cost for the first 3 metres = Rs 200 + Rs 50 = Rs 250
  • Cost for the first 4 metres = Rs 250 + Rs 50 = Rs 300

The sequence of costs is 150, 200, 250, 300, ...

The difference between consecutive terms is: 200 - 150 = 50, 250 - 200 = 50, 300 - 250 = 50. Since the difference is constant (d=50), this situation forms an AP.

(iv) Amount with Compound Interest:

The formula for the amount (A) after 'n' years with principal P, rate r is A = P(1 + r/100)^n.

  • Amount after 1 year = 10000(1 + 8/100)^1
  • Amount after 2 years = 10000(1 + 8/100)^2
  • Amount after 3 years = 10000(1 + 8/100)^3

The sequence of amounts is 10000(1.08), 10000(1.08)^2, 10000(1.08)^3, ...

The difference between the second and first term is 10000(1.08)^2 - 10000(1.08) = 10000(1.08)(1.08 - 1) = 10000(1.08)(0.08).

The difference between the third and second term is 10000(1.08)^3 - 10000(1.08)^2 = 10000(1.08)^2(1.08 - 1) = 10000(1.08)^2(0.08).

Since the differences are not constant, this situation does not form an AP.

Question 2

Write the first four terms of the Arithmetic Progression (AP) when the first term 'a' and the common difference 'd' are given as follows:

(i) a = 10, d = 10

(ii) a = -2, d = 0

(iii) a = 4, d = -3

(iv) a = -1, d = 1/2

(v) a = -1.25, d = -0.25

Solution:

The formula to find the nth term of an AP is a_n = a + (n-1)d, where 'a' is the first term and 'd' is the common difference. We can find the first four terms by substituting n=1, 2, 3, and 4.

(i) Given a = 10 and d = 10:

  • First term (a_1) = a = 10
  • Second term (a_2) = a + d = 10 + 10 = 20
  • Third term (a_3) = a + 2d = 10 + 2(10) = 10 + 20 = 30
  • Fourth term (a_4) = a + 3d = 10 + 3(10) = 10 + 30 = 40

The first four terms are 10, 20, 30, 40.

(ii) Given a = -2 and d = 0:

  • First term (a_1) = a = -2
  • Second term (a_2) = a + d = -2 + 0 = -2
  • Third term (a_3) = a + 2d = -2 + 2(0) = -2 + 0 = -2
  • Fourth term (a_4) = a + 3d = -2 + 3(0) = -2 + 0 = -2

The first four terms are -2, -2, -2, -2.

(iii) Given a = 4 and d = -3:

  • First term (a_1) = a = 4
  • Second term (a_2) = a + d = 4 + (-3) = 1
  • Third term (a_3) = a + 2d = 4 + 2(-3) = 4 - 6 = -2
  • Fourth term (a_4) = a + 3d = 4 + 3(-3) = 4 - 9 = -5

The first four terms are 4, 1, -2, -5.

(iv) Given a = -1 and d = 1/2:

  • First term (a_1) = a = -1
  • Second term (a_2) = a + d = -1 + 1/2 = -1/2
  • Third term (a_3) = a + 2d = -1 + 2(1/2) = -1 + 1 = 0
  • Fourth term (a_4) = a + 3d = -1 + 3(1/2) = -1 + 3/2 = 1/2

The first four terms are -1, -1/2, 0, 1/2.

(v) Given a = -1.25 and d = -0.25:

  • First term (a_1) = a = -1.25
  • Second term (a_2) = a + d = -1.25 + (-0.25) = -1.50
  • Third term (a_3) = a + 2d = -1.25 + 2(-0.25) = -1.25 - 0.50 = -1.75
  • Fourth term (a_4) = a + 3d = -1.25 + 3(-0.25) = -1.25 - 0.75 = -2.00

The first four terms are -1.25, -1.50, -1.75, -2.00.

Common mistakes

  • Confusing AP with other types of sequences (e.g., geometric progressions).
  • Incorrectly calculating the common difference.
  • Errors in applying the formula for subsequent terms.
  • Misinterpreting the conditions for forming an AP in real-world scenarios.

Revision tips

  • Focus on the definition of AP and the role of the common difference.
  • Practice identifying APs in various contexts, as shown in Question 1.
  • Work through all parts of Question 2 to solidify the calculation of terms.
  • Review the explanations for why certain situations do not form an AP.

Practice MCQs

Q1. Which of the following sequences is an Arithmetic Progression?

Q2. If the first term of an AP is 'a' and the common difference is 'd', what is the second term?

Q3. In which situation is the list of numbers NOT an Arithmetic Progression?

Q4. What is the common difference if the first term is 4 and the third term is -3?

Q5. If a = -2 and d = 0, what are the first four terms of the AP?

Frequently asked questions

What is an Arithmetic Progression (AP)?

An Arithmetic Progression is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference (d).

How do I know if a list of numbers forms an AP?

To check if a list of numbers forms an AP, calculate the difference between each pair of consecutive terms. If all these differences are the same, the list forms an AP.

What is the common difference in an AP?

The common difference (d) is the constant value added to each term to get the next term in an Arithmetic Progression.

How are the first four terms of an AP calculated?

Given the first term 'a' and the common difference 'd', the first four terms are: a, a+d, a+2d, and a+3d.

Does compound interest form an AP?

No, the amount of money in an account with compound interest does not form an AP because the increase each year is a percentage of the current amount, leading to a varying difference between terms.

What is the significance of 'd=0' in an AP?

If the common difference 'd' is 0, all the terms in the AP are identical to the first term.

Important links

Content reviewed by the NCERT Help team. Editorial Team and update policy

NCERT Solutions PDF PDF on NCERT Help. URL unchanged for search indexing.