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 $R s .150$ and rises by $R s . 20$ for each succeeding metre
(ii) The amount of air present in the cylinder when a vacuum pump removes each time $\frac{1}{4}$ 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
$a_{1} = 150 (i . e ., f i r s t t e r m o f A . P)$

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

cost of digging nth meter is given by.
$a_{n} = a_{1} + (n - 1) d$
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 $a_{1} = x$ then first time vacuum pump removes $\frac{1}{4}$ of the remaining air in cylinder.

hence remaining air is $a_{2} = x - \frac{x}{4} = \frac{3 x}{4}$

again second time vacuum pump removes $\frac{1}{4}$ of the remaining air in cylinder.
hence remaining air is $a_{3} = \frac{3 x}{4} - \frac{3 x}{16} = \frac{9 x}{16}$

the common difference in first and second term is
$d = a_{2} - a_{1} = \frac{3 x}{4} - x$
$d_{1} = \frac{- x}{4}$

the common difference in second and third term is
$d = a_{3} - a_{3} = \frac{9 x}{16} - \frac{3 x}{4}$
$d_{2} = \frac{- 3 x}{4}$
$s i n c e, d_{1} \neq d_{2}$
hence their is not a constant common difference
hence it is not an A.P

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Similar questions

Q. 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

\frac{1}{4}

of their remaining in the cylinder.

Q

In which of the following situations, does the list of numbers involved make an arithmetic progression, and why?
(i)The amount of air present in a cylinder when a vacuum pump removes ( $\frac{1}{4})^{t h}$ of the air remaining in the cylinder at a time.
(ii)The cost of digging a well after every meter of digging, when it costs Rs 150 for the first meter and rises by Rs 50 for each subsequent meter.

In which of the following situations, does the list of numbers involved make as 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.

In which of the following situations, does the list of numbers involved make an arithmetic progression, and why?
(i) The taxi fare after each $k m$ when the fare is $R s .15$ for the first $k m$ and $R s .8$ for each additional $k m$ .
(ii) The amount of air present in a cylinder when a vacuum pump removes $\frac{1}{4}$ 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 $R s .150$ for the first metre and rises by $R s .50$ for each subsequent metre.
(iv) The amount of money in the account every year, when $R s .10000$ is deposited at compound interest at $8 %$ per annum.

Q. In the following situations does the list of numbers involved make an arithmetic progression and why?
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.

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