Question

Find the r th term of an A.P., the sum of whose first n terms is 3n² + 2n. 

Answer 1

\[\text { Let a and d be the first term and the common difference of the given A . P . , respectively }\]

\[\text { As, } S_n = 3 n^2 + 2n\]

\[\text { So, } a = S_1 = 3 \times 1^2 + 2 \times 1 = 3 + 2 = 5 \text { and }\]

\[ S_2 = 3 \times 2^2 + 2 \times 2 = 12 + 4 = 16\]

\[ \Rightarrow a + a_2 = 16\]

\[ \Rightarrow a + a + d = 16\]

\[ \Rightarrow 2a + d = 16\]

\[ \Rightarrow 2 \times 5 + d = 16\]

\[ \Rightarrow d = 16 - 10\]

\[ \Rightarrow d = 6\]

\[\text { Now }, \]

\[ a_r = a + \left( r - 1 \right)d\]

\[ = 5 + \left( r - 1 \right) \times 6\]

\[ = 5 + 6r - 6\]

\[ \therefore a_r = 6r - 1\]