Question

Question 1 (ii)
In which of the following situations does the list of numbers involved make an arithmetic progression and why?
(ii) The amount of air present in a cylinder when a vacuum pump removes $\frac{1}{4}$ of the air remaining in the cylinder at a time.

Answer 1

Let us assume, initial quantity of air = V

Therefore, quantity removed in the first step = V $\times$ $\frac{1}{4}$

Quantity remaining after the first step

= $V-\frac{V}{4}$ = $\frac{3V}{4}$

Quantity removed in the second step = $\frac{3V}{4}$ $\times$ $\frac{1}{4}$

= $\frac{3V}{16}$

Quantity remaining after the second step

= $\frac{3V}{4}$ - $\frac{3V}{16}$ = $\frac{9V}{16}$

Quantity remaining after each step follows the trend:

$V,\frac{3V}{4},\frac{9V}{16},...$

Here each subsequent term is not obtained by adding a fixed number to the previous term.

Hence, it is not an AP.