Exercise 1 - Chapter 5 - Arithmetic Progressions, Class 10, Mathematics PDF Download

Arithmetic Progression

Exercise 1

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 fare is Rs. 15 for the first km and Rs. 8 for each additional km.

Solution: There is a common difference d and except first term each term can be obtained by using the common difference. This is arithmetic progression.

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

Solution: This is not an arithmetic progression. The difference will keep on decreasing as per remaining volume, so it cannot be said as a common difference.

(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.

Solution: This is arithmetic progression. The common difference is 50

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

Solution: This is not an arithmetic progression, as every year the value of interest will increase, because it is a compound interest.

Question: 2. Write first four terms of the AP, when the first term a and the common difference d are given as follows:

(i) a = 10 d = 10

Solution: n₁ = a + d(1-1) = 10 + 0 = 0

n₂ = a + d(2-1) = 10+10=20

n₃ = a + d(3-1) = 10 + 20 = 30

n₄ = a + d(4-1) = 10+30 = 40

(ii) a = -2 d = 0

Solution: Every term in this AP will be equal to -2 because d is 0.

(iii) a = 4, d = - 3 

Solution: n₁ = 4

n₂ = 4 – 3(1) = 1

n₃ = 4 – 3(2) = -2

n₄ = 4-3(3) = -5

(iv) a = -1 d = ½

Solution: n₁ = -1

n₂ = -1 + ½ = -1/2

n₃ = -1/2 + ½ = 0

n₄ = 0 + ½ = ½

(v) a = -1.25 d = -0.25

Solution:- n₁ = -1.25

n₂ = -1.5

n₃ = -1.75

n₄ = -2

3. For the following APs, write the first term and the common difference:

(i) 3, 1, -1, -3, …..

Solution: First Term a = 3

n₂ – n₁ = 1 – 3 = -2

n₃ – n₂ = -1 – 1 = -2

So, common difference = -2

(ii) -5, -1, 3, 7, ……

Solution: First Term a = -5

Common Difference = 4

(iii) 1/3, 5/3, 9/3, 13/3, …….

(iv) 0.6, 1.7, 2.8, 3.9, …….

Solution: First term a = 0.6

n₂-n₁ = 1.7 – 0.6 = 1.1

n₃ – n₂ = 2.8 – 1.7 = 1.1

So, common difference = 1.1

4. Which of the following are APs? If they form an AP, find the common difference d and write three more terms.

(i) 2, 4, 8, 16

Solution: This is not an AP as numbers are getting doubled

(ii) 2, 5/2, 3, 7/2, ……

Solution: This is an AP and common difference is 0.5

(iii) -1.2, -3.2 , -5.2, -7.2, …..

Solution: Common Difference is -2

(iv) -10, -6, -2, 2, …

Solution: Common Difference is 4

(vi) 0.2, 0.22, 0.222, 0.2222, ……

Solution: This is not an AP

(vii) 0, -4, -8, -12, …..

Solution: Common Difference is -4

(viii) -1/2, -1/2, -1/2, -1/2, …..

Solution: Common difference is 0

(ix) 1,3, 9, 27,……

Solution: This is not an AP

(x) a, 2a, 3a, 4a, …….

Solution: Common difference is 1a

(xi) a, a², a³, a⁴, ….

Solution: This is not an AP

Solution: This is not an AP

(xiii) 12, 52, 72, 73, ……

Solution: The series can be written as follows:

1, 25, 49, 73

Common difference is 24

(xv) 12, 32, 52, 72, ……

Solution: After rewriting we get

1, 9, 25, 49

As the difference is increasing at every term so this is not an AP.