<s>[INST] Find the value of the the term The sequence $a_1,a_2,a_3,\ldots$ satisfies $a_1=1$, $a_2=2$, and $$a_{n+2}=\frac2{a_{n+1}}+a_n\;;$$ find the value of 
$$\frac{a_{2012}2^{2009}}{2011}$$
 [/INST] As lab bhattacharjee shows, $a_{n+2}a_{n+1}=2n+2$, and thus,
$$
\begin{align}
\frac{a_{n+2}}{a_n}
&=\frac{a_{n+2}}{a_{n+1}}\frac{a_{n+1}}{a_n}\\
&=\frac{n+1}{n}
\end{align}
$$
Therefore,
$$
\begin{align}
\frac{a_{2n}}{a_2}
&=\frac{2n-1}{2n-2}\frac{2n-3}{2n-4}\cdots\frac32\\
&=\color{#C00000}{\frac1{4^{n-1}}\frac{(2n-1)!}{(n-1)!(n-1)!}}\\
&=\frac{n}{2^{2n-1}}\binom{2n}{n}
\end{align}
$$
and since $a_2=2$,
$$
a_{2n}=\frac{n}{4^{n-1}}\binom{2n}{n}
$$
I believe this verifies lab's hint (in red).
</s>