Question

Let there be an A.P. with first term 'a', common difference 'd'. If a_n denotes in n^th term and S_n the sum of first n terms, find.

\[k, \text{ if }  S_n = 3 n^2 + 5n \text{ and }  a_k = 164\]

 

Answer 1

\[a_k = S_k - S_{k - 1} \]
\[ \Rightarrow 164 = \left( 3 k^2 + 5k \right) - \left( 3 \left( k - 1 \right)^2 + 5\left( k - 1 \right) \right)\]
\[ \Rightarrow 164 = 3 k^2 + 5k - 3 k^2 + 6k - 3 - 5k + 5\]
\[ \Rightarrow 164 = 6k + 2\]
\[ \Rightarrow 6k = 162\]
\[ \Rightarrow k = 27\]
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