Question

In which of the following situations, the sequence of numbers formed will form an A.P.?

(i) The cost of digging a well for the first metre is Rs 150 and rises by Rs 20 for each succeeding metre.

(ii) The amount of air present in the cylinder when a vacuum pump removes each time 1/4 of their remaining in the cylinder. 

(iii) Divya deposited Rs 1000 at compound interest at the rate of 10% per annum. The amount at the end of first year, second year, third year, …, and so on.

Answer 1

(i) Given,

Cost of digging a well for the first meter (c₁) = Rs.150.

And, the cost rises by Rs.20 for each succeeding meter

Then,

Cost of digging for the second meter (c₂) = Rs.150 + Rs 20 = Rs 170

Cost of digging for the third meter (c₃) = Rs.170 + Rs 20 = Rs 210

Hence, its clearly seen that the costs of digging a well for different lengths are 150, 170, 190, 210, ….

Evidently, this series is in A∙P.

With first term (a) = 150, common difference (d) = 20

(ii) Given,

Let the initial volume of air in a cylinder be V liters each time 3^th/4 of air in a remaining i.e

1 -1/4

First time, the air in cylinder is V.

Second time, the air in cylinder is 3/4 V.

Third time, the air in cylinder is (3/4)² V.

Thus, series is V, 3/4 V, (3/4)² V,(3/4)³ V, ….

Hence, the above series is not a A.P.

(iii) Given,

Divya deposited Rs 1000 at compound interest of 10% p.a

So, the amount at the end of first year is = 1000 + 0.1(1000) = Rs 1100

And, the amount at the end of second year is = 1100 + 0.1(1100) = Rs 1210

And, the amount at the end of third year is = 1210 + 0.1(1210) = Rs 1331

Cleary, these amounts 1100, 1210 and 1331 are not in an A.P since the difference between them is not the same.