Question 1

We will check if the difference between consecutive terms is constant for each situation.

Step 1 — Taxi fare calculation

First km fare ($a_1$) = ₹ 15
Fare for each additional km = ₹ 8

Fares for subsequent kilometers: $a_1 = 15$ $a_2 = 15 + 8 = 23$ $a_3 = 23 + 8 = 31$ $a_4 = 31 + 8 = 39$

Differences between consecutive terms: $a_2 - a_1 = 23 - 15 = 8$ $a_3 - a_2 = 31 - 23 = 8$ $a_4 - a_3 = 39 - 31 = 8$

Since the difference is constant, this forms an Arithmetic Progression (AP).

This is an arithmetic progression.

Step 2 — Air in cylinder calculation

Let the initial volume of air be $V$.
Each stroke removes $\frac{1}{4}$ of the remaining air, leaving $1 - \frac{1}{4} = \frac{3}{4}$ of the previous volume.

Air volumes after successive strokes: $a_1 = V$ $a_2 = V \times \frac{3}{4} = \frac{3}{4}V$ $a_3 = \frac{3}{4}V \times \frac{3}{4} = \frac{9}{16}V$ $a_4 = \frac{9}{16}V \times \frac{3}{4} = \frac{27}{64}V$

Differences between consecutive terms: $a_2 - a_1 = \frac{3}{4}V - V = -\frac{1}{4}V$ $a_3 - a_2 = \frac{9}{16}V - \frac{3}{4}V = -\frac{3}{16}V$

Since the differences are not constant, this does not form an AP.

This is not an arithmetic progression.

Step 3 — Cost of digging a well calculation

Cost for the first metre ($a_1$) = ₹ 150
Cost increase per subsequent metre = ₹ 50

Costs for subsequent metres: $a_1 = 150$ $a_2 = 150 + 50 = 200$ $a_3 = 200 + 50 = 250$ $a_4 = 250 + 50 = 300$

Differences between consecutive terms: $a_2 - a_1 = 200 - 150 = 50$ $a_3 - a_2 = 250 - 200 = 50$ $a_4 - a_3 = 300 - 250 = 50$

Since the difference is constant, this forms an AP.

This is an arithmetic progression.

Step 4 — Compound interest calculation

Principal ($P$) = ₹ 10000
Rate of interest ($r$) = 8% per annum

Amount after $n$ years is given by $A = P\left(1 + \frac{r}{100}\right)^n$.

Amounts at the end of each year: $a_1 = 10000$ $a_2 = 10000 \left(1 + \frac{8}{100}\right) = 10800$ $a_3 = 10000 \left(1 + \frac{8}{100}\right)^2 = 11664$ $a_4 = 10000 \left(1 + \frac{8}{100}\right)^3 = 12597.12$

Differences between consecutive terms: $a_2 - a_1 = 10800 - 10000 = 800$ $a_3 - a_2 = 11664 - 10800 = 864$

Since the differences are not constant, this does not form an AP.

This is not an arithmetic progression.

Answer

(i) The list of numbers forms an AP because the difference between consecutive terms is constant ($d = 8$). (ii) The list of numbers does not form an AP because the difference between consecutive terms is not constant. (iii) The list of numbers forms an AP because the difference between consecutive terms is constant ($d = 50$). (iv) The list of numbers does not form an AP because the difference between consecutive terms is not constant.