In which of the following situations,
Question:

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.

Solution:

(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