Problem $1.(30\%)$ Answer the following questions on inhomogeneous recursion.

$($a$)(15\%)$ Consider the following recursion:

$T(n)=\{\begin{array}{cc}3T(n-1)+n,& n1\\ 0,& n=0\end{array}$

Solve the recursion using the method introduced in class ${?}^1.$

$($b$)(15\%)$ Consider the following recursion:

$T(n)=\{\begin{array}{cc}4T(n-2)+2^n,& n2\\ 1,& n=1\\ 0,& n=0\end{array}$

Solve the recursion using the method introduced in class.