Question

In which of the following situations, does the list of numbers involved make an arithmetic progression, and why?
(i)The taxi fare after each km when the fare is ${\rm{Rs}}15$ for the first km and ${\rm{Rs}}8$ for each additional km.
(ii)The amount of air present in a cylinder when a vacuum pump removes $\dfrac{1}{4}$ of the air remaining in the cylinder at a time.

Answer 1

Here, we are required to show whether the numbers involved in the given situations make an Arithmetic Progression or not. We will try to make a mathematical representation of the situations. We will make a series of the numbers from the two situations. If the difference between each term and its preceding term turns out to be the same then that situation will be an Arithmetic Progression.

Complete step-by-step answer:
According to the question,
Taxi fare for the first km is ${\rm{Rs}}15$.
Let us consider this as the first term of a series.
Hence, first term, $a = 15$
Now, it is given that for each additional km, the taxi fare is ${\rm{Rs}}8$
Hence, taxi fare for 2 km $= \left( {15 + 8} \right) = {\rm{Rs}}23$
Also, taxi fare for 3 km will be $= \left( {23 + 8} \right) = {\rm{Rs}}31$
And so on…
Hence, the series is:
$15,23,31,...$
Now, in order to prove whether this series is an Arithmetic Progression (A.P.) or not, we will find the difference between each term and its preceding term.
Hence, difference between the second and the first term $= 23 - 15 = 8$
Also, difference between the third and the second term $= 31 - 23 = 8$
We can see that the difference between each term and its preceding term is the same.
Hence, the common difference, $d = 8$
Therefore, the list of numbers involved in this situation make an Arithmetic Progression or an A.P.
Let the volume of the cylinder be 16 liters (as this is divisible by 4)
Now, according to the question,
A vacuum pump removes $\dfrac{1}{4}$ of the air remaining in the cylinder at a time. So,
Air removed by pump on the first time $= \dfrac{1}{4} \times 16 = 4$ liters
Now we will find the air present after removal by subtracting air removed by pump for the first time from the volume of the cylinder. Therefore, we get
Air present after the removal $= 16 - 4 = 12$ liters
Now, air removed by the pump on second time $= \dfrac{1}{4} \times 12 = 3$ liters
Again we will find the air present after removal by subtracting air removed by pump for the first time from air present after removal for the first time. Therefore, we get
Air present after the removal $= 12 - 3 = 9$ liters
Therefore, the amount of air present in the cylinder after the removal by the pump makes a series:
$16,12,9,...$
Now, in order to prove whether this series is an Arithmetic Progression (A.P.) or not, we will find the difference between each term and its preceding term.
The difference between the second and the first term $= 12 - 16 = - 4$
Also, difference between the third and the second term $= 9 - 12 = - 3$
We can see that the difference between each term and its preceding term is not the same.
Therefore, the list of numbers involved in this situation does not make an Arithmetic Progression (A.P.)

Note: Arithmetic Progressions are the sequences in which the difference between the consecutive terms is constant. In order to find whether a given series is an Arithmetic Progression or not, we should keep in mind that the common difference remains the same for all the consecutive terms.
Here, we might make a mistake by finding the common ratio instead of common difference and hence, end but finding whether the series is in geometric progression.