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 ₹15 for the first km and ₹ 8 for each additional km.
(ii) The amount of air present in a cylinder when a vacuum pump removes 1/4 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 ₹ 150 for the first metre and rises by ₹ 50 for each subsequent metre.
(iv) The amount of money in the account every year, when ₹ 10000 is deposited at compound interest at 8 % per annum.

Solution:

An arithmetic progression is a sequence of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.

General form of an arithmetic progression is a, (a + d), (a + 2d), (a + 3d), .. Where a is the first term and d is the common difference.

(i) Taxi fare for 1 km = ₹ 15 (a₁)

Taxi fare for 2 km = 15 + 8 = ₹ 23 (a₂)

Taxi fare for 3 km = 15 + 8 +  8 = ₹ 31 (a₃)

(a₂) - (a₁) = ₹ (23 - 15) = ₹ 8

(a₃) - (a₂) = ₹ (31 - 23) = ₹ 8

We see that the difference is constant between the terms.

So, this forms an AP with the first term as 15 and the common difference 8.

(ii) Let the amount of air in the cylinder be x.

So a₁ = x

After first time removal, a₂ = x - (x/4) = 3x/4

After second time removal,

a₃ = (3x/4) - (1/4) (3x/4)

= (3x/4) - (3x/16)

= (12x - 3x)/16

= 9x/16

After third time removal

a₄ = (9x/16) - (1/4) (9x/16)

= (9x/16) - (9x/64)

= (36x - 9x)/64

= 27x/64

Now,

a₂ - a₁ = (3x/4) - x

= (3x - 4x) / 4

= -x/4

a₃ - a₂ = (9x/16) - (3x/4)

= (9x - 12x)/16

= -3x/16

(a₃ - a₂) ≠ (a₂ - a₁)

Since the difference between the terms is not the same, this is not forming an AP.

(iii) Cost of digging the well after 1 meter = ₹ 150 (a₁)

Cost of digging the well after 2 meters = ₹ 150 + 50 = ₹ 200 (a₂)

Cost of digging the well after 3 meters = ₹ 150 + 50 + 50 = ₹ 250 (a₃)

(a₂ - a₁) = 200 - 150 = 50

(a₃ - a₂) = 250 - 200 = 50

(a₂ - a₁) = (a₃ - a₂)

We see that the difference is constant between the terms.

So, this froms an AP with the first term as ₹ 150 and the common difference is ₹ 50.

(iv) Amount present when the sum is P at an interest rate of r % after n years is

A =  P(1 + r/100)ⁿ

P = 10000

r = 8%

For first year, n = 1, (a₁) = 10000 (1 + 8/100)

For second year, n = 2, (a₂) = 10000 (1 + 8/100)²

For third year, n = 3, (a₃) = 10000 (1 + 8/100)³

For fourth year, n = 4, (a₄) = 10000 (1 + 8/100)⁴

a₂ - a₁ = [10000 (1 + 8/100)²] - [10000 (1 + 8/100)]

= [10000 (1 + 8/100)] [(1 + 8/100) - 1)

=  [10000 (1 + 8/100)] (8/100) 

a₃ - a₂

= [10000 (1 + 8/100)³] - [10000 (1 + 8/100)²]

= [10000 (1 + 8/100)²] [1 + 8/100 - 1]

= [10000 (1 + 8/100)² ] (8/100)

(a₃ - a₂) ≠ (a₂ - a₁)

Since the difference between the terms is not the same thus, the amount will not form an AP.

☛ Check: NCERT Solutions for Class 10 Maths Chapter 5

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Video Solution:

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 ₹15 for the first km and ₹ 8 for each additional km. (ii) The amount of air present in a cylinder when a vacuum pump removes 1/4 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 ₹ 150 for the first metre and rises by ₹ 50 for each subsequent metre. (iv) The amount of money in the account every year, when ₹ 10000 is deposited at compound interest at 8 % per annum

NCERT Solutions Class 10 Maths Chapter 5 Exercise 5.1 Question 1

Summary:

(i) The taxi fare after each km when the fare is ₹15 for the first km and ₹ 8 for each additional km. This forms an AP (ii) The amount of air present in a cylinder when a vacuum pump removes 1/4 of the air remaining in the cylinder at a time. This does not form an AP(iii) The cost of digging a well after every metre of digging when it costs ₹ 150 for the first metre and rises by ₹ 50 for each subsequent metre. This forms an AP (iv) The amount of money in the account every year when ₹ 10000 is deposited at compound interest at 8 % per annum. This does not form an AP.

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