<s>[INST] Find $ S_{20} $ if $ \frac{ a_{n} +2}{2} = \sqrt{2S_{n}} $ for all integer n Define $ a_{n} $ is a sequence and all terms of $ a_{n} $ are positive. $ S_{n} $ is the summation of the first n terms. If $ \frac{ a_{n} +2}{2} = \sqrt{2S_{n}} $ for all integer n, then find $ S_{20} $
 [/INST] $S_n=a_1+a_2+\cdots+a_n=S_{n-1}+a_n=S_{n-1}+2(\sqrt{2S_n}-1)\Rightarrow$ $S_{19}+2(\sqrt{2S_{20}}-1)=S_{20}$. Also we have $S_{20}-S_{19}=a_{20}$. Adding these two we get 
$2\sqrt{2S_{20}}=a_{20}\Rightarrow S_{20}=\frac{a_{20}^2}{8}$.
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