$(5$ points$)$ Consider the recursion $u_t=3u_{t-1}-b.$

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

$($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 $u_t=3u_{t-1}-b$ to a geometric recursion of the form $v_t=v_{t-1}.$