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Question
Find an A.P. in which the sum of any number of terms is always three times the squared number of these terms.
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Solution
Given:
\[S_n = 3 n^2 \]
\[\text { For } n = 1, S_1 = 3 \times 1^2 = 3\]
\[\text { For } n = 2, S_2 = 3 \times 2^2 = 12\]
\[\text { For } n = 3, S_3 = 3 \times 3^2 = 27 \]
\[\text { and so on }\]
\[ \therefore S_1 = a_1 = 3\]
\[ a_2 = S_2 - S_1 = 12 - 3 = 9\]
\[ a_3 = S_3 - S_2 = 27 - 12 = 15\]
\[\text { and so on }\]
\[\text { Thus, the A . P . is } 3, 9, 15 . . . \]
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