In which of the following situations, does the list of numbers involved make an arithmetic progression, and why?
The taxi fare after each km when the fare is ₹ 15 for the first km and ₹ 8 for each additional km.
Yes, this forms an arithmetic progression. The taxi fare after each km is: ₹ 15, ₹ 23, ₹ 31, ₹ 39, ... The common difference = 23 - 15 = 31 - 23 = 39 - 31 = ₹ 8. Since each term is obtained by adding a fixed number (₹ 8) to the preceding term, it forms an AP.
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.
No, this does not form an arithmetic progression. If initial air is 1 unit, the amounts are: 1, 3/4, 9/16, ... The differences are: 3/4 - 1 = -1/4 and 9/16 - 3/4 = -3/16. Since -1/4 -3/16, the common difference is not fixed. Each term is obtained by multiplying by 3/4, not by adding a fixed number.
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.
Yes, this forms an arithmetic progression. The cost after each metre is: ₹ 150, ₹ 200, ₹ 250, ₹ 300, ... The common difference = 200 - 150 = 250 - 200 = 300 - 250 = ₹ 50. Since each term is obtained by adding a fixed number (₹ 50) to the preceding term, it forms an AP.
The amount of money in the account every year, when ₹ 10000 is deposited at compound interest at 8% per annum.
No, this does not form an arithmetic progression. The amounts after each year are: ₹ 10000, ₹ 10800, ₹ 11664, ... The differences are: 10800 - 10000 = 800 and 11664 - 10800 = 864. Since 800 864, the common difference is not fixed. In compound interest, the increase grows each year.
Explanation
According to the textbook context, an arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term. This fixed number is called the common difference. For situations (i) and (iii), the taxi fare and digging cost increase by a fixed amount each time (₹ 8 and ₹ 50 respectively), forming an AP. For situations (ii) and (iv), the changes are multiplicative (removing 1/4 of remaining air, compound interest), so the difference between consecutive terms varies, not forming an AP.