Question

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) \(2, \frac{5}{2},3,\frac{7}{2}\), . . . .
(iii) – 1.2, – 3.2, – 5.2, – 7.2, . . . .
(iv) – 10, – 6, – 2, 2, . . .
(v) 3, \(3 + \sqrt{2} , 3 + 3\sqrt{2} , 3 + 3 \sqrt{2}\) . . . .
(vi) 0.2, 0.22, 0.222, 0.2222, . . . .
(vii) 0, – 4, – 8, –12, . . . .
(viii) \(\frac{-1}{2}, \frac{-1}{2}, \frac{-1}{2}, \frac{-1}{2}\), . . . .
(ix) 1, 3, 9, 27, . . . .
(x) a, 2a, 3a, 4a, . . . .
(xi) a, \(a^2, a^3, a^4,\)  . . . .
(xii) \(\sqrt{2}, \sqrt{8} , \sqrt{18} , \sqrt {32}\) . . . .
(xiii) \(\sqrt {3}, \sqrt {6}, \sqrt {9} , \sqrt {12}\) . . . . .
(xiv) \(1^2 , 3^2 , 5^2 , 7^2\), . . . .
(xv) \(1^2 , 5^2, 7^2, 7^3\), . . . .

Answer 1

(i) The sequence 2, 4, 8, 16... is analyzed. The differences between consecutive terms are: \(a_2 - a_1 = 4 - 2 = 2\), \(a_3 - a_2 = 8 - 4 = 4\), and \(a_4 - a_3 = 16 - 8 = 8\). Since the difference \(a_{k+1} - a_k\) is not constant, these numbers do not form an Arithmetic Progression (A.P.).

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(ii) The sequence 2, \( \frac{5}{2}, 3, \frac{7}{2} \) is analyzed. The differences between consecutive terms are: \(a_2-a_1 = \frac{5}{2}-2 = \frac{1}{2}\), \(a_3 - a_2 = 3- \frac{5}{2} = \frac{1}{2}\), and \(a_4 - a_3 = \frac{7}{2} - 3 = \frac{1}{2}\). Since the difference \(a_{k+1} - a_k\) is constant, these numbers form an A.P. with a common difference d = \( \frac{1}{2} \). The next three terms are \(a_5 = \frac{7}{2} + \frac{1}{2} = 4\), \(a_6 = 4 + \frac{1}{2} = \frac {9}{2}\), and \(a_7 = \frac{9}{2} + \frac{1}{2} = 5\).

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(iii) The sequence - 1.2, - 3.2, - 5.2, - 7.2... is analyzed. The differences between consecutive terms are: \(a_2 - a_1 = ( - 3.2) - ( - 1.2) = - 2\), \(a_3 - a_2 = ( - 5.2) - ( - 3.2) = - 2\), and \(a_4 - a_3 = ( - 7.2) - ( - 5.2) = - 2\). Since the difference \(a_{k+1} - a_k\) is constant, these numbers form an A.P. with a common difference d = - 2. The next three terms are \(a_5 = - 7.2 - 2 = - 9.2\), \(a_6 = - 9.2 - 2 = - 11.2\), and \(a_7 = - 11.2 - 2 = - 13.2\).

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(iv) The sequence - 10, - 6, - 2, 2... is analyzed. The differences between consecutive terms are: \(a_2 - a_1 = ( - 6) - ( - 10) = 4\), \(a_3 - a_2 = ( - 2) - ( - 6) = 4\), and \(a_4 - a_3 = (2) - ( - 2) = 4\). Since the difference \(a_{k+1} - a_k\) is constant, these numbers form an A.P. with a common difference d = 4. The next three terms are \(a_5 = 2 + 4 = 6\), \(a_6 = 6 + 4 = 10\), and \(a_7 = 10 + 4 = 14\).

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(v) The sequence 3, \(3+\sqrt{2} , 3+ 2\sqrt{2} , 3 + 3 \sqrt{2}\), ..... is analyzed. The differences between consecutive terms are: \(a_2 - a_1 = 3 + \sqrt{2}-3 = \sqrt{2}\), \(a_3 - a_2 = 3 + 2 \sqrt{2} - 3 - \sqrt{2}= \sqrt{2}\), and \(a_4 - a_3 = 3 + 3 \sqrt{2} - 3 - 2 \sqrt{2}= \sqrt{2}\). Since the difference \(a_{k+1}-a_k\) is constant, these numbers form an A.P. with a common difference d = \(\sqrt{2}\). The next three terms are \(a_5 = 3 + 3 \sqrt{2} + \sqrt{2} = 3 + 4 \sqrt{2}\), \(a_6 = 3 + 4 \sqrt{2} + \sqrt{2} = 3 + 5 \sqrt{2}\), and \(a_7 = 3 + 5 \sqrt{2} + \sqrt{2} = 3 + 6 \sqrt{2}\).

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(vi) The sequence 0.2, 0.22, 0.222, 0.2222... is analyzed. The differences between consecutive terms are: \(a_2 - a_1 = 0.22 - 0.2 = 0.02\), \(a_3 - a_2 = 0.222 - 0.22 = 0.002\), and \(a_4 - a_3 = 0.2222 - 0.222 = 0.0002\). Since the difference \(a_{k+1} - a_k\) is not constant, these numbers do not form an A.P.

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(vii) The sequence 0, - 4, - 8, - 12... is analyzed. The differences between consecutive terms are: \(a_2-a_1 = ( - 4) - 0 = - 4\), \(a_3-a_2 = ( - 8) - ( - 4) = - 4\), and \(a_4 - a_3 = ( - 12) - ( - 8) = - 4\). Since the difference \(a_{k+1}-a_k\) is constant, these numbers form an A.P. with a common difference d = - 4. The next three terms are a+(5 - 1)d = -16, a+(6 - 1)d = -20, and a+(7 - 1)d = -24.

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(viii) This sequence is an A.P. with a common difference of 0. Therefore, the next three terms will be the same as the previous ones, i.e., \(-\frac{1}{2}\).

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(ix) The differences between the first three terms are 9 - 3 = 6 and 3 - 1 = 2. Since 6 ≠ 2, \(a_3 - a_2 eq a_2 - a_1\). Therefore, the given numbers do not form an A.P.

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(x) This sequence is an A.P. with a common difference d = 2a - a = a and the first term is a. The next three terms are a + (5 - 1)d = 5a, a + (6 - 1)d = 6a, and a + (7 - 1)d = 7a.

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(xi) This sequence is not an A.P. because the difference between consecutive terms is not constant.

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(xii) This sequence is an A.P. with a common difference d = \(\sqrt{2}\) and the first term a = \(\sqrt{2}\). The next three terms are a + (5 - 1)d = \(5\sqrt{2}=\sqrt{50}\), a + (6 - 1)d = \(\sqrt{72}\), and a + (7 - 1)d = \(\sqrt{98}\).

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(xiii) This sequence is not an A.P. because the difference between consecutive terms is not constant.

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(xiv) This sequence is not an A.P. because the difference between consecutive terms is not constant.

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(xv) This sequence is an A.P. with a common difference d = 5² - 1 = 24 and the first term a = 1. The next three terms are a + (5 - 1)d = 97, a + (6 - 1)d = 121, and a + (7 - 1)d = 145.