In which of the following situation, the sequence of numbers formed wi

1.	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 \(\frac{1}{4}\) of their remaining in the cylinder.
Answer» (i) The cost of digging a well for the first metre = Rs. 150 The cost of digging a well for the second metre = 150 + 20 = Rs. 170 The cost of digging a well for the third metre = 170 + 20 = Rs. 190 The cost of digging a well for the fourth metre = 190 + 20 = Rs. 210 Difference between first and second metre = 170 – 150 = 20 Difference between second and third metre = 190 – 170 = 20 Difference between third and fourth metre = 210 – 190 = 20 Since, all the differences are equal. Therefore, it’s an A.P. (ii) Let the amount of air present in the cylinder first time = x Amount of air present in the cylinder second time = x - \(\frac{x}{4}\) = \(\frac{3x}{4}\) Amount of air present in the cylinder third time = \(\frac{3x}{4}\) - \(\frac{3x}{16}\) = \(\frac{9x}{16}\) Difference in the amount of air present in the cylinder between first and the second time = \(\frac{3x}{4}\) – x = \(\frac{-x}{4}\) Difference in the amount of air present in the cylinder between second and the third time = \(\frac{9x}{16}\) - \(\frac{3x}{4}\) = \(\frac{-3x}{16}\) Since, the differences are unequal. Therefore, it’s not an A.P.

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 \(\frac{1}{4}\) of their remaining in the cylinder.

Answer»

(i) The cost of digging a well for the first metre = Rs. 150

The cost of digging a well for the second metre = 150 + 20 = Rs. 170

The cost of digging a well for the third metre = 170 + 20 = Rs. 190

The cost of digging a well for the fourth metre = 190 + 20 = Rs. 210

Difference between first and second metre = 170 – 150 = 20

Difference between second and third metre = 190 – 170 = 20

Difference between third and fourth metre = 210 – 190 = 20

Since, all the differences are equal.

Therefore, it’s an A.P.

(ii) Let the amount of air present in the cylinder first time = x

Amount of air present in the cylinder second time = x - \(\frac{x}{4}\) = \(\frac{3x}{4}\)

Amount of air present in the cylinder third time = \(\frac{3x}{4}\) - \(\frac{3x}{16}\) = \(\frac{9x}{16}\)

Difference in the amount of air present in the cylinder between first and the second time = \(\frac{3x}{4}\) – x = \(\frac{-x}{4}\)

Difference in the amount of air present in the cylinder between second and the third time = \(\frac{9x}{16}\) - \(\frac{3x}{4}\) = \(\frac{-3x}{16}\)