Question

In which of the following situations, does the list of numbers involved make an arithmetic progression, and why?
(i)The amount of air present in a cylinder when a vacuum pump removes ($\frac{1}{4}{)}^{th}$ of the air remaining in the cylinder at a time.
(ii)The cost of digging a well after every meter of digging, when it costs Rs 150 for the first meter and rises by Rs 50 for each subsequent meter.

Answer 1

(i) Let amount of air initially present in a cylinder = V

Amount of air left after pumping out air by vacuum pump = V − $\frac{V}{4}$= $\frac{3}{4}$ V

Amount of air left when vacuum pump again pumps out air

= $\frac{3}{4}$V − [$\frac{1}{4}$ × $\frac{3}{4}$V] = $\frac{3}{4}$V−$\frac{3}{16}$V = $\frac{9}{16}$V

So, the sequence we get is like V, $\frac{3}{4}$V, $\frac{9}{16}$V....

Checking for difference between consecutive terms.

$\frac{3}{4}$V−V = −$\frac{V}{4}$

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

Difference between consecutive terms is not constant. Therefore, it is not an arithmetic progression.

(ii)

Cost of digging 1 meter of well = Rs 150

Cost of digging 2 meters of well = 150+50= Rs 200

Cost of digging 3 meters of well = 200+50 = Rs 250

Therefore, we get a sequence of the form 150, 200, 250....

It is an arithmetic progression because difference between any two consecutive terms is constant.

(200-150 =250-200 = 50)