In which of the following situations, do the lists of numbers involved form an AP? Give reasons for your answers.
(i) The fee charged from a student every month by a school for the 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 the account of Varun at the end of every year when Rs. 1000 is deposited at simple interest of `10%` per annum.
(iv) The number of bacteria in a certain food item after each second, when they double in every second.

Explanation

(i) The fee charged from a student every month by a school for the whole session is
` 400, 400, 400, 400, …`
Which form an AP, with common difference (d) `=400-400=0`
(ii) The fee charged month by a school from I to XII is
`250,(250+50),(250+2xx50),(250+3xx50), ...`
i.e, `250,300,350,400, ...`
which form an AP, with common difference (d) `=300-250=50`
(iii) Simple interest `=("Principal" xx "Rate" xx "Time")/(100)`
` " " =(1000xx10xx1)/(100)=100`
So, the amount of money in the account of Varun at the end of every year is
`1000,(1000+100xx1),(1000+100xx2),(1000+100xx3), ...`
i.e., `1000,1100,1200,1300, ...`
which form an AP, with common difference (d) `=1100-1000=100`
(iv) Let the number of bacteria in a certain food `= x`
Since, they double in every sceond.
` :. x,2x,2(2x),2(2*2*x), ...`
i.e., `x,2x,4x,8x, ...`
Now, let ` " " t_(1)=x,t_(2)=2x,t_(3)=4x` and `t_(4)=8x`
`t_(2)-t_(1)=2x-x=x`
`t_(3)-t_(2)=4x-2x=2x`
`t_(4)-t_(3)=8x-4x=4x`
Since, the difference between each successive term is not same. So, the list does form an AP.