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 ₹ 15 for the first km and ₹ 8 for each additional km.
(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 ₹ 150 for the first metre and rises by ₹ 50 for each subsequent metre.
(iv) The amount of money in the account every year, when ₹ 10000 is deposited at compound interest at 8% per annum.
(i) Yes. The total fare after each km is \(15, 23, 31, \ldots\), which forms an AP because each succeeding term is obtained by adding 8 to the preceding term.
(ii) No. The volumes of air are \(V, \frac{3V}{4}, \left(\frac{3}{4}\right)^2 V, \ldots\), which form a geometric progression, not an AP.
(iii) Yes. The costs are \(150, 200, 250, \ldots\), which form an AP with common difference 50.
(iv) No. The amounts are \(10000\left(1 + \frac{8}{100}\right), 10000\left(1 + \frac{8}{100}\right)^2, 10000\left(1 + \frac{8}{100}\right)^3, \ldots\), which form a geometric progression, not an AP.
Write first four terms of the AP, when the first term \(a\) and the common difference \(d\) are given as follows:
(i) \(a = 10,\ d = 10\)
(ii) \(a = -2,\ d = 0\)
(iii) \(a = 4,\ d = -3\)
(iv) \(a = -1,\ d = \frac{1}{2}\)
(v) \(a = -1.25,\ d = 0.25\)
(i) \(10, 20, 30, 40\)
(ii) \(-2, -2, -2, -2\)
(iii) \(4, 1, -2, -5\)
(iv) \(-1, -\tfrac{1}{2}, 0, \tfrac{1}{2}\)
(v) \(-1.25, -1.50, -1.75, -2.00\)
For the following APs, write the first term and the common difference:
(i) \(3, 1, -1, -3, \ldots\)
(ii) \(-5, -1, 3, 7, \ldots\)
(iii) \(\tfrac{1}{3}, \tfrac{5}{3}, \tfrac{9}{3}, \tfrac{13}{3}, \ldots\)
(iv) \(0.6, 1.7, 2.8, 3.9, \ldots\)
(i) First term \(a = 3\), common difference \(d = -2\).
(ii) First term \(a = -5\), common difference \(d = 4\).
(iii) First term \(a = \tfrac{1}{3}\), common difference \(d = \tfrac{4}{3}\).
(iv) First term \(a = 0.6\), common difference \(d = 1.1\).
Which of the following are APs? If they form an AP, find the common difference \(d\) and write three more terms.
(i) \(2, 4, 8, 16, \ldots\)
(ii) \(2, \tfrac{5}{2}, 3, \tfrac{7}{2}, \ldots\)
(iii) \(-1.2, -3.2, -5.2, -7.2, \ldots\)
(iv) \(-10, -6, -2, 2, \ldots\)
(v) \(3, 3 + \sqrt{2}, 3 + 2\sqrt{2}, 3 + 3\sqrt{2}, \ldots\)
(vi) \(0.2, 0.22, 0.222, 0.2222, \ldots\)
(vii) \(0, -4, -8, -12, \ldots\)
(viii) \(\tfrac{1}{2}, -\tfrac{1}{2}, \tfrac{1}{2}, -\tfrac{1}{2}, \ldots\)
(ix) \(1, 3, 9, 27, \ldots\)
(x) \(a, 2a, 3a, 4a, \ldots\)
(xi) \(a, a^2, a^3, a^4, \ldots\)
(xii) \(\sqrt{2}, \sqrt{8}, \sqrt{18}, \sqrt{32}, \ldots\)
(xiii) \(\sqrt{3}, \sqrt{6}, \sqrt{9}, \sqrt{12}, \ldots\)
(xiv) \(1^2, 3^2, 5^2, 7^2, \ldots\)
(xv) \(1^2, 5^2, 7^2, 73, \ldots\)
(i) No.
(ii) Yes, it is an AP with \(d = \tfrac{1}{2}\); next three terms are \(4, \tfrac{9}{2}, 5\).
(iii) Yes, it is an AP with \(d = -2\); next three terms are \(-9.2, -11.2, -13.2\).
(iv) Yes, it is an AP with \(d = 4\); next three terms are \(6, 10, 14\).
(v) Yes, it is an AP with \(d = \sqrt{2}\); next three terms are \(3 + 4\sqrt{2}, 3 + 5\sqrt{2}, 3 + 6\sqrt{2}\).
(vi) No.
(vii) Yes, it is an AP with \(d = -4\); next three terms are \(-16, -20, -24\).
(viii) No.
(ix) No.
(x) Yes, it is an AP with \(d = a\); next three terms are \(5a, 6a, 7a\).
(xi) No.
(xii) Yes, it is an AP with \(d = \sqrt{2}\); next three terms are \(\sqrt{50}, \sqrt{72}, \sqrt{98}\).
(xiii) No.
(xiv) No.
(xv) Yes, it is an AP with \(d = 24\); next three terms are \(97, 121, 145\).