(Q21).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
1
4
 th of the air remaining in the cylinder at a time.

Let the amount of air initially present in the cylinder be 1024 lit.

First it removes
1
4
 th of the volume

i.e.,
1
4
 × =

∴ Remaining air present in the cylinder = 1024 - =

At second time it removes
1
4
 th of

i.e.,
1
4
 × =

∴ Remaining air in the cylinder = – =

Again at third time it removes
1
4
 th of 576

i.e.,
1
4
 × =

Remaining air in the cylinder = 576 – =

i.e., the volume of the air present in the cylinder after 1st, 2nd, 3rd,… times is 1024, , , , …..

Here, a2 – a1 = – = –

a3 – a2 = – = –

a4 – a3 = – = – .

Thus the difference between any two successive terms is not equal to a number.

∴ The given situation doesn’t show an A.P.