$(5$ points$)$ Consider the recursion ut $=3$ut$1$ b$.$

$($a$)$ Find the value of b such that the equilibrium solution of this recursion is $=500.$

$($b$)$ Using the value of b you found in question $($a$),$ make a change of variables and show how this change transforms the affine geometric recursion ut $=3$ut$1$ b to a geometric recursion of the form vt $=\backslash$lambda vt$1.$