Which of the following lists of numbers form an A.P. ? If they form an A.P., find the common difference d and write the next three terms: (i) 4, 10, 16, 22, .... (ii) -2, 2, -2, 2, .... (iii) 2, 4, 8, 16, .... (iv) (v) -10, -6, -2, 2, ... (vi) 1², 3², 5², 7², ....

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(i) Given, 4, 10, 16, 22, .... Here, a₂ - a₁ = 10 - 4 = 6, a₃ - a₂ = 16 - 10 = 6, a₄ - a₃ = 22 - 16 = 6 i.e. any term - preceding term = 6, a fixed number. Hence, the given list of numbers forms an A.P. with common difference = d = 6. For the next three terms, we have: a₅ = a₄ + d = 22 + 6 = 28, a₆ = a₅ + d = 28 + 6 = 34, a₇ = a₆ + d = 34 + 6 = 40. Hence, the given series is in A.P. with common difference d = 6 and the next three terms : 28, 34, 40. (ii) Given, -2, 2, -2, 2, .... Here, a₂ - a₁ = 2 - (-2) = 4, a₃ - a₂ = -2 - 2 = -4, a₄ - a₃ = 2 - (-2) = 4 ⇒ a₂ - a₁ = a₄ - a₃ ≠ a₃ - a₂. Thus the difference of any term from its preceding term is not a fixed number. Hence, the given series does not form an A.P. (iii) Given, 2, 4, 8, 16, .... Here, a₂ - a₁ = 4 - 2 = 2, a₃ - a₂ = 8 - 4 = 4, a₄ - a₃ = 16 - 8 = 8 ⇒ a₂ - a₁ ≠ a₃ - a₂ ≠ a₄ - a₃. Thus the difference of any term from its preceding term is not a fixed number. Hence, the given series does not form an A.P. (iv) Given, Here, a₂ - a₁ = , a₃ - a₂ = , a₄ - a₃ = i.e. any term - preceding term = , a fixed number. Hence, the given list of numbers forms an A.P. with common difference = d = . For the next three terms, we have: a₅ = a₄ + d = , a₆ = a₅ + d = , a₇ = a₆ + d = . Hence, the given series is in A.P. with common difference d = and the next three terms : 4, , 5. (v) Given, -10, -6, -2, 2, ... Here, a₂ - a₁ = -6 - (-10) = 4, a₃ - a₂ = -2 - (-6) = 4, a₄ - a₃ = 2 - (-2) = 4 i.e. any term - preceding term = 4, a fixed number. Hence, the given list of numbers forms an A.P. with common difference = d = 4. For the next three terms, we have: a₅ = a₄ + d = 2 + 4 = 6, a₆ = a₅ + d = 6 + 4 = 10, a₇ = a₆ + d = 10 + 4 = 14. Hence, the given series is in A.P. with common difference d = 4 and the next three terms : 6, 10, 14. (vi) Given, 1², 3², 5², 7², .... or, 1, 9, 25, 49 .... Here, a₂ - a₁ = 9 - 1 = 8, a₃ - a₂ = 25 - 9 = 16, a₄ - a₃ = 49 - 25 = 24 ⇒ a₂ - a₁ ≠ a₃ - a₂ ≠ a₄ - a₃. Thus the difference of any term from its preceding term is not a fixed number. Hence, the given series does not form an A.P.

Mathematics

AP GP

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Mathematics

AP GP

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