Determine the sum $T=a_0+a_1+a_2+...+a_{2012}$ Let ${a_n}$, $n \ge 0$ be a sequence of positive real numbers, given by
$a_0=1$ and
$a_m<a_n$ for all $m,n \in \mathbb{N}, m<n$ with 
$a_n=\sqrt{a_{n+1}a_{n-1}}+1$ and $4\sqrt{a_n}=a_{n+1}-a_{n-1}$ for all $n \in \mathbb{N}, n\neq 0$.
Help me, determining the sum $T=a_0+a_1+a_2+...+a_{2012}$.

Take $n=1$ in both the equations, you will get, $a_1=\sqrt {a_2}+1$ and $4\sqrt {a_1}=a_2-1$ (here i substituted $a_0=1$), solving these equations gives $a_1=4=2^2$ and $a_2=9=3^2$. Now you can calculate $a_3$ by putting $n=2$ in any one of the given equations(the equations are consistent(you can check by using both to obtain answer)) which gives you $a_3=16=4^2$. Following this manner, you will get further terms as $25,36,49\cdots {2013}^2$. Therefore sum is $1^2+2^2+3^2+\cdots +{2013}^2=2721031819$
