Q) 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 $\frac{\mathrm{1/}}{4}$ of the air remaining in the cylinder at a time.

Ans: Let’s start from making a sequence of the air amount in the cylinder after every stroke and then check if the sequence makes an AP or not.

Let’s consider the initial amount of air in the cylinder is “X”.

Clearly this becomes our 1^st term,

Next, it is given that the vacuum pump removes of the air present,

hence, air removed by pump in the 1^st stroke =

and the balance air in the cylinder will be =

This becomes our 2^nd term,

Next, air removed by pump in the 2^nd stroke = x

and the balance air in the cylinder will be =

This becomes our 3^rd term,

Similarly, air removed by pump in the 4^th stroke = x

and the balance air in the cylinder will be =

This becomes our 4^th term,

Thus, the sequence starts to emerge as

Next, let’s check the difference between 2 sets of consecutive terms:

Difference between 1^st two terms:

Similarly, difference between next two terms:

and, difference between next two terms:

Here, since the difference between any two consecutive terms is not equal, hence the sequence formed is not an AP

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