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$\begin{align*} &A_n : \left \{ 2,4,8,{\color{red}{\bf 14}},22,{\color{red}{\bf 32}},44... \right \} \\ &\text{Consecutive term differences are in AP : } 2,4,6,8... \\ &\text{Define } S_{{\color{red}{\bf k}}} = \text{Sum up to }{{\color{red}{\bf k}}}\text{th term in the AP series 2,4,6,8,10...} \\ &\text{Where common diff} = {{\color{red}{\bf 2}}} \\ \\ \hline \\ \\ &\text{Now ,} \\ &\Rightarrow A_n = 2 + S_{n-1} \\ &\text{For example :} \\ &A_4= {\color{red}{\bf 14}} = 2 + S_{4-1} =2 + S_{3}= 2 + \frac{3}{2}.\left [ 2*2+(3-1)*2 \right ]=14 \\ &A_6= {\color{red}{\bf 32}} = 2 + S_{6-1} =2 + S_{5}= 2 + \frac{5}{2}.\left [ 2*2+(5-1)*2 \right ]=32 \\ &\Rightarrow A_{{\color{red}{\bf 99}}} = 2 + S_{99-1} =2 + S_{98}= 2 + \frac{98}{2}.\left [ 2*2+(98-1)*2 \right ]=\bf 9704 \\ \end{align*}$
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