Question

In which of the following situations, does the list of numbers involved make as arithmetic progression and why?
(1) The taxi fare after each km when the fare is Rs 15 for the first km and Rs 8 for each additional km.
(2) The amount of air present in a cylinder when a vacuum pump removes \(\frac{1}{4}\) of the air remaining in the cylinder at a time.
(3) 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.
(4) The amount of money in the account every year, when Rs 10000 is deposited at compound interest at 8% per annum.

Answer 1

(i) Observation: Taxi fare for the 1st km is 15. For the first 2 km, it's 15 + 8 = 23. For the first 3 km, it's 23 + 8 = 31. For the first 4 km, it's 31 + 8 = 39. The sequence 15, 23, 31, 39... forms an A.P. as each term is 8 more than the preceding one.

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(ii) Let the initial volume of air in a cylinder be V liters. Each stroke of the vacuum pump removes \( \frac{1}{4} \) of the remaining air. This means \( 1 - \frac{1}{4} = \frac{3}{4} \) of the air remains after each stroke. The volumes will be \( V, \frac{3}{4}V, (\frac{3}{4})^2V, (\frac{3}{4})^3V \).... The adjacent terms in this series do not have a constant difference. Therefore, this sequence is not an A.P.

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(iii) Cost of digging the first meter is 150. For the first 2 meters, it's 150 + 50 = 200. For the first 3 meters, it's 200 + 50 = 250. For the first 4 meters, it's 250 + 50 = 300. The sequence 150, 200, 250, 300... forms an A.P. because each term is 50 more than the previous term.

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(iv) With Rs P deposited at an annual compound interest rate of r% for n years, the amount after n years will be \( P (1 +\frac{r}{100})^n \). For an initial deposit of 10000 at 8% interest, the amounts after each year are: \( 10000(1 + \frac{8}{100}) \), \( 10000 (1 + \frac{8}{100})^2 \), \( 10000 (1 + \frac{8}{100})^3 \), \( 10000 (1 + \frac{8}{100})^4 \), ...... The adjacent terms in this series do not have the same difference. Therefore, this is not an A.P.