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 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 removes of 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.

Answer 1

(i) Rent for	Rent for	Rent for
1st km	2nd km	3rd km
Rs. 15	Rs. 8	Rs. 8

d = a₂ – a₁ = 8 – 15 = -7 

d = a₃ – a₂ = 8 – 8 = 0 

Here ’d’ is not constant. 

Hence it does not form an AP. 

(ii) Let the amount of air present in a cylinder be ‘x’. 

1. \(\frac{1}{4}\) of 1 = \(\frac{1}{4}\) Amount of air goes out 

1 - \(\frac{1}{4}\) = \(\frac{3}{4}\) Air remaining 

2. \(\frac{1}{4}\) of \(\frac{3}{4}\) = \(\frac{3}{16}\) Amount of air goes out 

\(\frac{3}{4}\) - \(\frac{3}{16}\) = \(\frac{9}{16}\) Air remaining 

3.\(\frac{1}{4}\) of \(\frac{9}{16}\) = \(\frac{9}{64}\) Amount of air goes out 

\(\frac{9}{16}\) - \(\frac{9}{64}\) = \(\frac{27}{64}\) Air remaining

Here the value of ‘d’ is not constant. 

∴ The given list of numbers does not form an A.P. 

(iii) Cost of Digging for the

d = a₂ – a₁ = 200 – 150 = 50 

d = a₃ – a₂ = 250 – 200 = 50 

Here value of ‘d’ is constant. 

This forms an AP. 

(iv) Amount of money deposited in the 

d = a₂ – a₁ = 11,664 – 10.800 = Rs. 864 

d = a₃ – a₂ = 12,597,12 – 11.664 = Rs. 933.12 

Here value of ‘d’ is not constant. 

∴ The given list of numbers does not form an AP.