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 $\dfrac{1}{4}$ 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.

For the following APs, write the first term and the common difference:

(i) 3, 1, -1, -3, …….

(ii) -5, -1, 3, 7, …….

(iii) $\dfrac{1}{3}, \dfrac{5}{3}, \dfrac{9}{3}, \dfrac{13}{3}$, …….

(iv) 0.6, 1.7, 2.8, 3.9, ……..

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, ….

(ii) 2, $\dfrac{5}{2}, 3, \dfrac{7}{2}, ……$

(iii) -1.2, -3.2, -5.2, -7.2, ……..

(iv) -10, -6, -2, 2, ………

(v) 3, 3 + $\sqrt{2}, 3 + 2\sqrt{2}, 3 + 3\sqrt{2}$, ……….

(vi) 0.2, 0.22, 0.222, 0.2222, ……….

(vii) 0, -4, -8, -12, ……..

(viii) $-\dfrac{1}{2}, -\dfrac{1}{2}, -\dfrac{1}{2}, -\dfrac{1}{2}, ………..$

(ix) 1, 3, 9, 27, ……..

(x) a, 2a, 3a, 4a, ……….

(xi) a, a², a³, a⁴, ………..

(xii) $\sqrt{2}, \sqrt{8}, \sqrt{18}, \sqrt{32},$ ………..

(xiii) $\sqrt{3}, \sqrt{6}, \sqrt{9}, \sqrt{12},$ ………..

(xiv) 1², 3², 5², 7², ……

(xv) 1², 5², 7², 73, ……

Fill in the blanks in the following table, given that a is the first term, d the common difference and a_n the nth term of the AP :