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 (\(d=\sqrt{3}\)).
Step 1: Recall the definition of an AP.
A sequence is called an Arithmetic Progression (AP) if the difference between any two consecutive terms is always the same. This difference is called the common difference, denoted by \(d\).
(i) \(-1, -1, -1, -1, \ldots\)
All terms are the same. So the difference between any two consecutive terms is \(-1 - (-1) = 0\).
Since the difference is always 0, it is an AP with \(d = 0\).
(ii) \(0, 2, 0, 2, \ldots\)
Find differences:
The differences are not the same (sometimes 2, sometimes -2). So it is not an AP.
(iii) \(1, 1, 2, 2, 3, 3, \ldots\)
Find differences:
The differences are changing (0, 1, 0, 1,...). So it is not an AP.
(iv) \(11, 22, 33, \ldots\)
Find differences:
The difference is always 11. So it is an AP with \(d = 11\).
(v) \(\tfrac{1}{2}, \tfrac{1}{3}, \tfrac{1}{4}, \ldots\)
Find differences:
The differences are not the same (\(-\tfrac{1}{6}, -\tfrac{1}{12}, ...\)). So it is not an AP.
(vi) \(2, 2^2, 2^3, 2^4, \ldots\)
This sequence is: \(2, 4, 8, 16, \ldots\).
Find differences:
The differences are not the same. This is actually a Geometric Progression (terms are multiplied by 2 each time). So it is not an AP.
(vii) \(\sqrt{3}, \sqrt{12}, \sqrt{27}, \sqrt{48}, \ldots\)
Simplify square roots:
So the sequence becomes: \(\sqrt{3}, 2\sqrt{3}, 3\sqrt{3}, 4\sqrt{3}, \ldots\).
Find differences:
The difference is always \(\sqrt{3}\). So it is an AP with \(d = \sqrt{3}\).