How to compute the recursive sequence with square? Consider a sequence defined by
$x_{n+1}^2=ax_n^2+bx_n+c$,where $a,b,c&gt;0$ and $x_n &gt;0$,
is there any general method to compute the exact form of $x_n$?
Thank you!
This doesn't define a sequence because $x_{n+1}$ could be
$+\sqrt{a x_n^2+b x_n+c}$
or
$-\sqrt{a x_n^2+b x_n+c}$.
Let's say you set $a=b=c=1$ and $x_0 = 1$, then the relation says
$x_1^2 = 1\cdot 1^2 + 1\cdot 1+ 1 = 3$.
This doesn't define $x_1$, it could be that $x_1 = \sqrt{3}$ or $x_1 = -\sqrt{3}$.
Do you have any information about the sign of $x_n$?
If you know $x_{n+1}$ then you can use the quadratic formula to find $x_n$.