(i) The taxi fare after each km when the fare is Rs. 15 for the first km and Rs. 8 for each additional km.
Solution:
We can write the given condition as;
Taxi fare for 1 km = 15
Taxi fare for first 2 km = 15 + 8 = 23
Taxi fare for first 3 km = 23 + 8 = 31
Taxi fare for first 4 km = 31 + 8 = 39
And so on……
Thus, 15, 23, 31, 39 … forms an A.P. because every next term is 8 more than the preceding term.
(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.
Solution:
Let the volume of air in a cylinder, initially, be V litres.
In each stroke, the vacuum pump removes 1/4th of the air remaining in the cylinder at a time. Or we can say that after every stroke, 1 – 1/4 = 3/4th part of the air will remain.
Therefore, volumes will be V, 3V/4 , (3V/4)2 , (3V/4)3…and so on
Clearly, we can see here, that the adjacent terms of this series do not have the common difference between them. Therefore, this series is not an A.P.
(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:
We can write the given condition as;
The cost of digging a well for the first metre = Rs.150
The cost of digging a well for the first 2 metres = Rs.150+50 = Rs.200
The cost of digging a well for the first 3 metres = Rs.200+50 = Rs.250
The cost of digging a well for the first 4 metres =Rs.250+50 = Rs.300
And so on.
Clearly, 150, 200, 250, 300, … form an A.P. with a common difference of 50 between each term.
(iv) The amount of money in the account every year, when Rs. 10000 is deposited at compound interest at 8% per annum.
Solution:
We know that if Rs. P is deposited at r% compound interest per annum for n years, the amount of money will be:
P(1 + r/100)n
Therefore, after each year, the amount of money will be;
10000(1 + 8/100), 10000(1 + 8/100)2, 10000(1 + 8/100)3……
Clearly, the terms of this series do not have a common difference between them. Therefore, this is not an A.P.