In which of the following situation, the sequence of numbers formed will form an A.P.?
(i) The cost of digging a well for the first metre is Rs. 150 and rises by Rs.20 for each succeeding metre.
(ii)The mount of air present in the cylinder when a vacuum pump removes each time 1/4 of their remaining in the cylinder.
Answers
(i) The sequence of numbers formed will form an A.P.
(ii) The sequence of numbers formed will not form an A.P.
Given,
(i) The cost of digging a well for the first metre is Rs. 150 and rises by Rs.20 for each succeeding metre.
Given,
The cost of digging a well for the first metre is Rs. 150
∴ a = 150
rises by Rs.20 for each succeeding metre.
⇒ 150, 150+20, 150+20+20, ................
⇒ 150, 170, 190,............
⇒ d = 170 - 150 = 190 - 170 = 20
∴ d = 20
Therefore, the series is given by, 150, 170, 190,............ with a common difference of 20.
Hence this situation represents the AP series
(ii)The mount of air present in the cylinder when a vacuum pump removes each time 1/4 of their remaining in the cylinder.
Let the amount of air present before any removal be 16 litres.
(chosen 16 as it is a multiple of 4, since given 1/4)
First time removal = 1/4 × 16 = 4
Air remaining= 16 - 4 = 12
Second time removal = 1/4 × 12 = 3
Air remaining= 12 - 3 = 9
Second time removal = 1/4 × 9 = 9/4
The series is 16, 12, 9, ...........
d = 12 - 16 ≠ 9 - 12
The series has no common difference.
Hence this situation does not represent the AP series