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Question

In which of the following situations, 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 amount of air present in the cylinder when a vacuum pump removes each time 14 of their remaining in the cylinder.

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Solution

1) Given, the cost of digging a well for the first meter is 150 and rises by Rs. 20.

hence for first meter (n=1)the cost is given by
a1=150 (i.e.,first term of A.P)

and rises by Rs. 20 which is the common difference of A.P.
hence d=20

cost of digging nth meter is given by.
an=a1+(n1)d
hence this problem is an A.P 150,170,190,210,.......
because their is a constant common difference


2) let initially the cylinder has amount of air a1=x then first time vacuum pump removes 14 of the remaining air in cylinder.

hence remaining air is a2=xx4=3x4

again second time vacuum pump removes 14 of the remaining air in cylinder.
hence remaining air is a3=3x43x16=9x16

the common difference in first and second term is
d=a2a1=3x4x
d1=x4

the common difference in second and third term is
d=a3a3=9x163x4
d2=3x4
since,d1d2
hence their is not a constant common difference
hence it is not an A.P

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