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Question
Find the r th term of an A.P., the sum of whose first n terms is 3n2 + 2n.
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Solution
\[\text { Let a and d be the first term and the common difference of the given A . P . , respectively }\]
\[\text { As, } S_n = 3 n^2 + 2n\]
\[\text { So, } a = S_1 = 3 \times 1^2 + 2 \times 1 = 3 + 2 = 5 \text { and }\]
\[ S_2 = 3 \times 2^2 + 2 \times 2 = 12 + 4 = 16\]
\[ \Rightarrow a + a_2 = 16\]
\[ \Rightarrow a + a + d = 16\]
\[ \Rightarrow 2a + d = 16\]
\[ \Rightarrow 2 \times 5 + d = 16\]
\[ \Rightarrow d = 16 - 10\]
\[ \Rightarrow d = 6\]
\[\text { Now }, \]
\[ a_r = a + \left( r - 1 \right)d\]
\[ = 5 + \left( r - 1 \right) \times 6\]
\[ = 5 + 6r - 6\]
\[ \therefore a_r = 6r - 1\]
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