<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] HINT:
$$a_{n+2}=\frac2{a_{n+1}}+a_n\iff a_{n+2}a_{n+1}=a_{n+1}a_n+2$$
$$\implies a_{n+2}a_{n+1}=2n+a_2a_1=2n+2\ \ \ \ (1)$$
$$\text{Replacing }n\text{ with } n-1,\implies a_{n+1}a_n=2n$$
$$\implies a_{n+2}a_{n+1}\cdot2n=a_{n+1}a_n(2n+2)$$
$$\implies a_{n+2} =a_n\frac{(n+1)}n$$
$$n=2m-2\implies a_{2m} =a_{2m-2}\frac{(2m-1)}{2m-2}$$
$$ a_{2m} =\frac{(2m-1)(2m-3)\cdots3}{(2m-2)(2m-4)\cdots4}a_2$$
$$\text{Now,} \frac{(2m-1)(2m-3)\cdots3}{(2m-2)(2m-4)\cdots4}=2\frac{(2m-1)(2m-2)\cdots4\cdot3\cdot2\cdot1}{2^{m-1}\cdot (m-1)! ((2m-2)\cdot4\cdot2)}=\frac{(2m-1)!}{2^{2m-2} \{(m-1)!\}^2}$$ and $a_2=2$
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