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 Rs 15 for the first km and Rs 8 for each additional km.

(ii) The amount of air present in a cylinder when a vacuum pump removesof the air remaining in the cylinder at a time.

(iii) The cost of digging a well after every metre of digging, when it costs Rs 150 for the first metre and rises by Rs 50 for each subsequent metre.

(iv) The amount of money in the account every year, when Rs 10000 is deposited at compound interest at 8 % per annum

For a sequence to be AP, the difference of consecutive terms should remain constant and that is called the common difference of the AP
(i) Fare for 1^st km = Rs. 15

Fare for 2^nd km = Fare of first km + Additional fare for 1 km
                       =  Rs. 15 + 8
                       = Rs 23

Fare for 3^rd km = Fare of first km + Fare of additional second km + Fare of additional third km
                       = Rs. 23 + 8
                       = Rs 31

Fare of n km = 15 + 8 x (n - 1) 
( We multiplied by n - 1 because first km was fixed and for rest we are adding additional fare.

In this, each subsequent term is obtained by adding a fixed number (8) to the previous term.

Hence, it is an AP

(ii) Let us take initial quantity of air = 1

Hence, quantity of air removed in first step = 1/4

Remaining quantity after 1^st step

Quantity removed after 2^nd step = Quantity removed in first step x initial quantity

Remaining quantity after 2^nd step would be: 

After second step the difference between second and first and first and initial step is not the same, hence

Here a fixed number is not added to each subsequent term.

Hence, it is not an AP

(iii) Cost of digging of 1st meter = Rs 150

Cost of digging of second meter = cost of digging of first meter + cost of digging additional meter

Cost of digging of 2nd meter = 150 + 50

= Rs 200

Cost of digging of third meter = Cost of digging of first meter + cost of digging of second meter + cost of digging of third meter

Cost of digging of 3rd meter = 200 + 50

= Rs 250

Here, each subsequent term is obtained by adding a fixed number (50) to the previous term.

Hence, it is an AP.

(iv)Amount in the beginning = Rs. 10000

Interest at the end of 1st year at rate of 8%

= 10000 x 8%

= 800

Hence, amount at the end of 1^styear

= 10000 + 800

= 10800

Now the interest will be made at the principal taken as amount of first year, Hence

Interest at the end of 2^nd year at rate of 8%

= 10800 x 8%

= 864

Thus, amount at the end of 2^ndyear

= 10800 + 864 = 11664

Since, each subsequent term is not obtained by adding a unique number to the previous term; hence, it is not an AP