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# In which of the following situations, does the list of numbers involved make an arithmetic progression and why?

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.

Given:

The amount of air present in the cylinder when a vacuum pump removes each time $\frac{1}{4}$ of the remaining in the cylinder.

To do:

We have to check whether the sequence formed by the above information is an A.P.

Solution:

Let the air present n the cylinder be $1$. The amount of air removed the first time $=1\times\frac{1}{4}=\frac{1}{4}$

The amount of air remaining after removing the air first time $=1-\frac{1}{4}=\frac{1\times4-1}{4}=\frac{3}{4}$

The amount of air removed the second time $=\frac{3}{4}\times\frac{1}{4}=\frac{3}{16}$

The amount of air remaining after removing the air second time $=\frac{3}{4}-\frac{3}{16}=\frac{4\times3-3}{16}=\frac{9}{16}$

The amount of air removed the third time $=\frac{9}{16}\times\frac{1}{4}=\frac{9}{64}$

The amount of air remaining after removing the air third time $=\frac{9}{16}-\frac{9}{64}=\frac{9\times4-9}{64}=\frac{27}{64}$

The amount of air present in the cylinder when a vacuum pump removes each time $\frac{1}{4}$ of the remaining in the cylinder forms the below sequence

$1, \frac{3}{4}, \frac{9}{16}, \frac{27}{64}, ........$

$\frac{3}{4}-1=\frac{-1}{4}$, $\frac{9}{16}-\frac{3}{4}=\frac{-3}{16}$

$\frac{-1}{4}≠\frac{-3}{16}$

Therefore, the sequence of the numbers does not form an A.P.

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