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 \(97, 121, 145\).
How to check whether a sequence is an AP?
Subtract consecutive terms. If the difference is the same throughout, the sequence is an AP. That constant value is the common difference \(d\). If the differences vary, it is not an AP.
Differences: 4 − 2 = 2, 8 − 4 = 4, 16 − 8 = 8 → not constant → Not an AP.
Differences: \(\tfrac{5}{2} - 2 = \tfrac{1}{2}\), \(3 - \tfrac{5}{2} = \tfrac{1}{2}\), \(\tfrac{7}{2} - 3 = \tfrac{1}{2}\) → constant → AP with d = \(\tfrac{1}{2}\).
Next terms: add \(\tfrac{1}{2}\) repeatedly → \(4, \tfrac{9}{2}, 5\).
Differences: −3.2 − (−1.2) = −2, −5.2 − (−3.2) = −2 → constant → AP with d = −2.
Next terms: −9.2, −11.2, −13.2.
Differences: −6 − (−10) = 4, −2 − (−6) = 4 → constant → AP with d = 4.
Next terms: 6, 10, 14.
Differences: \((3 + \sqrt{2}) - 3 = \sqrt{2}\), \((3 + 2\sqrt{2}) - (3 + \sqrt{2}) = \sqrt{2}\) → constant → AP with d = \(\sqrt{2}\).
Next terms: \(3 + 4\sqrt{2}, 3 + 5\sqrt{2}, 3 + 6\sqrt{2}\).
Differences change each time → Not an AP.
Differences: −4, −4, −4 → constant → AP with d = −4.
Next terms: −16, −20, −24.
Differences: −1, +1, −1 → not constant → Not an AP.
Ratios are constant (geometric), but differences are not → Not an AP.
Differences: 2a − a = a, 3a − 2a = a → constant → AP with d = a.
Next terms: 5a, 6a, 7a.
Differences vary → Not an AP.
Simplify terms: \(\sqrt{8} = 2\sqrt{2}\), \(\sqrt{18} = 3\sqrt{2}\), \(\sqrt{32} = 4\sqrt{2}\).
The sequence becomes: \(\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}\) → clearly an AP with difference \(\sqrt{2}\).
Next terms: \(5\sqrt{2} = \sqrt{50}, 6\sqrt{2} = \sqrt{72}, 7\sqrt{2} = \sqrt{98}\).
Differences are not constant → Not an AP.
Sequence: 1, 9, 25, 49 → differences not constant → Not an AP.
Sequence: 1, 25, 49, 73 → differences: 24, 24, 24 → constant → AP with d = 24.
Next terms: 97, 121, 145.