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 ₹ 400.
(ii) The fee charged every month by a school from classes I to XII, When the monthly fee for class I is ₹ 250 and it increase by ₹ 50 for the next higher
class.
(iii) The amount of money in the account of Varun at the end of every year when ₹ 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.
(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 + 2 x 50), (250 + 3 x 50),…
i.e., 250,300,350,400,…
which form an AP, with common difference (d) = 300 – 250 = 50
(iii) Simple interest $=\frac{\text { Principal } \times \text { Rate } \times \text { Time }}{100}$
$=\frac{1000 \times 10 \times 1}{100}=100$
So, the amount of money in the account of Varun at the end of every year is
$1000,(1000+100 \times 1),(1000+100 \times 2),(1000+100 \times 3), \ldots$
i.e., $1000,1100,1200,1300, \ldots$
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 second.
$\therefore \quad x, 2 x, 2(2 x), 2(2 \cdot 2 \cdot x) \ldots$
i.e., $x, 2 x, 4 x, 8 x, \ldots$
Now, let $t_{1}=x, t_{2}=2 x, t_{3}=4 x$ and $t_{4}=8 x$
$t_{2}-t_{1}=2 x-x=x$
$t_{3}-t_{2}=4 x-2 x=2 x$
$t_{4}-t_{3}=8 x-4 x=4 x$
Since, the difference between each successive term is not same, So, the list does form an AP