CBSE Class 10 Mathematics NCERT Solutions: Chapter 5 Arithmetic Progression

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NCERT Solutions PDF Class 10 PDF

This chapter provides comprehensive NCERT Solutions for Class 10 Mathematics, focusing on Arithmetic Progressions (AP). Students will learn to identify whether a given sequence of numbers forms an arithmetic progression by checking for a constant common difference between consecutive terms. The solutions cover various scenarios, including real-world applications like taxi fares and digging costs, and abstract mathematical sequences. Key concepts include defining the first term (a) and the common difference (d), and using these to generate the first few terms of an AP. The chapter also lays the groundwork for understanding the general term and the sum of terms in an AP, crucial for exam preparation and building a strong foundation in sequences and series.

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BoardCBSE
ClassClass 10
SubjectMathematics
Session2026
LanguageEnglish
TypeNCERT Solutions
ChapterChapter 5: Arithmetic Progression

Chapter summary

Chapter 5 on Arithmetic Progressions for CBSE Class 10 Mathematics introduces the concept of sequences where the difference between consecutive terms is constant. This solution set covers identifying APs in various contexts, such as fare calculations and cost estimations. It details how to find the first four terms of an AP given the first term and common difference, reinforcing the fundamental definition of an AP.

Learning outcomes

  • Understand the definition of an Arithmetic Progression (AP).
  • Identify whether a given sequence is an AP by checking the common difference.
  • Determine the first four terms of an AP when the first term and common difference are provided.
  • Analyze real-world scenarios to determine if they represent an AP.
  • Differentiate between sequences that are APs and those that are not.

Topics covered

Paper topics

  • Definition of Arithmetic Progression (AP)
  • Identifying APs
  • Common Difference (d)
  • First Term (a)
  • Generating terms of an AP
  • Real-world applications of APs
  • Distinguishing APs from other sequences

Important topics

  • Definition and identification of AP
  • Calculating the common difference
  • Finding the first few terms of an AP
  • Real-world problem analysis for APs

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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 situation forms an Arithmetic Progression (AP), we need to check if the difference between consecutive terms is constant. This constant difference is called the common difference (d).

(i) Taxi Fare:

The fare for the first km is Rs 15.

The fare for the first 2 km is Rs 15 + Rs 8 = Rs 23.

The fare for the first 3 km is Rs 23 + Rs 8 = Rs 31.

The fare for the first 4 km is 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 common difference is constant (d=8), this situation forms an AP.

(ii) Air in a Cylinder:

Let the initial volume of air be V.

After the first removal, the remaining air is V - (1/4)V = (3/4)V.

After the second removal, the remaining air is (3/4)V - (1/4)(3/4)V = (3/4)V * (1 - 1/4) = (3/4)V * (3/4) = (3/4)^2 V.

After the third removal, the remaining air is (3/4)^2 V - (1/4)(3/4)^2 V = (3/4)^2 V * (3/4) = (3/4)^3 V.

The sequence of remaining 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, (3/4)^2 V - (3/4)V = -(1/4)(3/4)V. Since the differences are not constant (they depend on the term itself), this situation does not form an AP.

(iii) Cost of Digging a Well:

The cost for the first metre is Rs 150.

The cost for the first 2 metres is Rs 150 + Rs 50 = Rs 200.

The cost for the first 3 metres is Rs 200 + Rs 50 = Rs 250.

The cost for the first 4 metres is 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 common difference is constant (d=50), this situation forms an AP.

(iv) Amount with Compound Interest:

The initial deposit is Rs 10000.

After 1 year, the amount is $10000(1 + \frac{8}{100})^1$.

After 2 years, the amount is $10000(1 + \frac{8}{100})^2$.

After 3 years, the amount is $10000(1 + \frac{8}{100})^3$.

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

The difference between consecutive terms is $10000(1.08)^2 - 10000(1.08) = 10000(1.08)(1.08 - 1)$ and $10000(1.08)^3 - 10000(1.08)^2 = 10000(1.08)^2(1.08 - 1)$. Since the differences are not constant (they increase with each term), 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:
  1. a = 10, d = 10
  2. a = -2, d = 0
  3. a = 4, d = -3
  4. a = -1, d = \frac{1}{2}
  5. a = -1.25, d = -0.25
Solution:

The general form of an Arithmetic Progression is $a, a+d, a+2d, a+3d, \dots$, where 'a' is the first term and 'd' is the common difference. We need to find the first four terms for each given pair of 'a' and 'd'.

(i) Given: a = 10, 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, 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, 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, d = \frac{1}{2}

First term ($a_1$) = a = -1

Second term ($a_2$) = a + d = -1 + \frac{1}{2} = -\frac{1}{2}

Third term ($a_3$) = a + 2d = -1 + 2(\frac{1}{2}) = -1 + 1 = 0

Fourth term ($a_4$) = a + 3d = -1 + 3(\frac{1}{2}) = -1 + \frac{3}{2} = \frac{1}{2}

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

(v) Given: a = -1.25, 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

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

Revision tips

  • Focus on the definition of common difference (d) for every AP problem.
  • Practice identifying APs in diverse word problems.
  • Ensure you can correctly calculate the next terms using 'a' and 'd'.
  • Review the examples thoroughly to understand the reasoning behind classifying sequences as AP or not.

Practice MCQs

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

Q2. If the first term (a) of an AP is 5 and the common difference (d) is -3, 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 -1.25 and the second term is -1.50?

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 identify if a list of numbers is an AP?

To identify an AP, calculate the difference between each pair of consecutive terms. If all these differences are the same, the list is an AP.

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, a+3d.

Does compound interest form an AP?

No, the amount of money with compound interest does not form an AP because the increase each year is a percentage of the current amount, not a fixed amount.

What does a common difference of 0 mean in an AP?

A common difference of 0 means that all the terms in the AP are the same as the first term.

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