Which of the following form an AP? Justify your answer.
(i) \(-1, -1, -1, -1, \ldots\)
(ii) \(0, 2, 0, 2, \ldots\)
(iii) \(1, 1, 2, 2, 3, 3, \ldots\)
(iv) \(11, 22, 33, \ldots\)
(v) \(\dfrac{1}{2},\; \dfrac{1}{3},\; \dfrac{1}{4},\; \ldots\)
(vi) \(2, 2^2, 2^3, 2^4, \ldots\)
(vii) \(\sqrt{3},\; \sqrt{12},\; \sqrt{27},\; \sqrt{48},\; \ldots\)
(i) Yes (\(d=0\)); (ii) No; (iii) No; (iv) Yes (\(d=11\)); (v) No; (vi) No; (vii) Yes (\(d=\sqrt{3}\)).
Is it true that \(-1,\; -\dfrac{3}{2},\; -2,\; -\dfrac{5}{2},\; \ldots\) forms an AP because \(a_2-a_1=a_3-a_2\)? Justify.
True.
For the AP \(-3,-7,-11,\ldots\), can we find \(a_{30}-a_{20}\) directly without actually finding \(a_{30}\) and \(a_{20}\)? Explain.
Yes. \(a_{30}-a_{20}=(30-20)\,d=10(-4)=-40\).
Two APs have the same common difference. The first term of one AP is \(2\) and that of the other is \(7\). Show that the difference between their 10th terms equals the difference between their 21st terms (indeed, between any two corresponding terms). Why?
The difference is constant and equals \(2-7=-5\) for every corresponding term.
Is \(0\) a term of the AP \(31, 28, 25, \ldots\)? Justify your answer.
No.
The taxi fare after each km, when the fare is Rs 15 for the first km and Rs 8 for each additional km, does not form an AP as the total fare (in Rs) after each km is \(15, 8, 8, 8, \ldots\). Is this statement true? Give reasons.
False. The total fare after each km is \(15, 23, 31, 39, \ldots\), which is an AP with \(a=15\), \(d=8\).
In which of the following situations do the lists of numbers form an AP? Give reasons.
(i) The fee charged every month by a school for a whole session when the monthly fee is Rs 400.
(ii) The fee charged every month by a school from Classes I to XII, when the monthly fee for Class I is Rs 250, and it increases by Rs 50 for the next higher class.
(iii) The amount of money in Varun’s account at the end of every year when Rs 1000 is deposited at simple interest of 10% p.a.
(iv) The number of bacteria after each second when they double every second.
(i) Yes (\(d=0\)); (ii) Yes (\(d=50\)); (iii) Yes (\(d=100\)); (iv) No (geometric, not arithmetic).
Decide whether each expression can be the \(n\)th term of an AP. Justify.
(i) \(2n-3\)
(ii) \(3n^2+5\)
(iii) \(1+n+n^2\)
(i) Yes; (ii) No; (iii) No.