In which of the following situations do the lists of numbers form an AP? Give reasons.
(i) The fee charged every month by a school for a whole session when the monthly fee is Rs 400.
(ii) The fee charged every month by a school from Classes I to XII, when the monthly fee for Class I is Rs 250, and it increases by Rs 50 for the next higher class.
(iii) The amount of money in Varun’s account at the end of every year when Rs 1000 is deposited at simple interest of 10% p.a.
(iv) The number of bacteria after each second when they double every second.
(i) Yes (\(d=0\)); (ii) Yes (\(d=50\)); (iii) Yes (\(d=100\)); (iv) No (geometric, not arithmetic).
(i) The fee is the same every month: 400, 400, 400, ...
Here, the difference between any two months is 0. Since the difference is the same each time (0), this is an AP with common difference \(d=0\).
(ii) The fees are: 250 for Class I, 300 for Class II, 350 for Class III, and so on.
So the list is: 250, 300, 350, ...
The difference between each term is \(300 - 250 = 50\), \(350 - 300 = 50\), and so on. Since the difference is always the same (50), this is an AP with common difference \(d=50\).
(iii) Simple interest formula is \(A = P(1 + rt)\).
Here, \(P=1000\), \(r = 10\% = 0.1\), \(t\) is time in years.
For 1 year: \(1000(1 + 0.1 \times 1) = 1100\).
For 2 years: \(1000(1 + 0.1 \times 2) = 1200\).
For 3 years: \(1000(1 + 0.1 \times 3) = 1300\).
So the sequence is 1100, 1200, 1300, ... Each year it increases by 100. This is an AP with common difference \(d=100\).
(iv) The bacteria double every second.
Starting from 2, we get 2, 4, 8, 16, ...
Here, the change is not by addition but by multiplication (each time multiply by 2). This is called a Geometric Progression (GP), not an AP.