**(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*, 3*V*/4 , (3*V*/4)^{2} , (3*V*/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.