Let's analyze the sequences formed in the given situations to determine whether they form an arithmetic progression (A.P.).
An arithmetic progression (A.P.) is a sequence of numbers in which the difference between consecutive terms is constant.
Situation (ii): The amount of air present in the cylinder when a vacuum pump removes each time [tex]\(\frac{1}{4}\)[/tex] of the remaining air in the cylinder.
Let's denote the initial amount of air in the cylinder as [tex]\(A_0\)[/tex].
1. Initial amount of air: [tex]\(A_0\)[/tex]
2. After removing [tex]\(\frac{1}{4}\)[/tex] of the air, the amount of air left: [tex]\(A_1 = A_0 - \frac{1}{4} A_0 = \frac{3}{4} A_0\)[/tex]
3. After another removal of [tex]\(\frac{1}{4}\)[/tex] of the remaining air, the amount left: [tex]\(A_2 = A_1 - \frac{1}{4} A_1 = \frac{3}{4} A_1 = \left(\frac{3}{4}\right)^2 A_0\)[/tex]
4. After another removal: [tex]\(A_3 = A_2 - \frac{1}{4} A_2 = \frac{3}{4} A_2 = \left(\frac{3}{4}\right)^3 A_0\)[/tex]
The sequence of the amounts of air is [tex]\(A_0, \frac{3}{4} A_0, \left(\frac{3}{4}\right)^2 A_0, \left(\frac{3}{4}\right)^3 A_0, \ldots\)[/tex], which is a geometric progression (G.P.) with common ratio [tex]\(\frac{3}{4}\)[/tex]. Since the difference between terms is not constant, this sequence does not form an A.P.
Situation (iii): Divya deposited ₹1000 at compound interest at the rate of 10% per annum. The amount at the end of the first year, second year, third year, ..., and so on.
Let's denote the initial deposit as [tex]\(P = ₹1000\)[/tex].
1. After the first year, the amount [tex]\(A_1\)[/tex] is given by: [tex]\(A_1 = P \left(1 + \frac{r}{100}\right) = 1000 \left(1 + \frac{10}{100}\right) = 1000 \times 1.10 = ₹1100\)[/tex]
2. After the second year, the amount [tex]\(A_2\)[/tex] is given by: [tex]\(A_2 = A_1 \left(1 + \frac{r}{100}\right) = 1100 \times 1.10 = ₹1210\)[/tex]
3. After the third year, the amount [tex]\(A_3\)[/tex] is given by: [tex]\(A_3 = A_2 \left(1 + \frac{r}{100}\right) = 1210 \times 1.10 = ₹1331\)[/tex]
The sequence of amounts is ₹1000, ₹1100, ₹1210, ₹1331, ..., which follows the compound interest formula. The amounts do not increase by a constant difference, as each term is 110% of the previous term. Therefore, this sequence does not form an A.P.
Conclusion
In summary, neither of the given situations forms an arithmetic progression (A.P.). Both examples involve sequences where the difference between consecutive terms is not constant.
