$(18$ points$)$ Derive asymptotic upper bounds for $T(n)$ in the following recurrences. Make your bounds as tight as possible. Do not use the master theorem. Derive your solutions from scratch. Show all your steps in order to gain full credits $($either using the repeated expansion method or the recursion tree method$).$ You may assume that $n$ is a power of $4$ for $($a$),n$ is a power of $3$ for $($b$),$ and $n$ is always of a power of $5$ and $4$ for $($c$).$

$($a$)T(1)=1$;$T(n)=8T(\frac{n}{4})+n$ for $n>1.$

$($b$)T(1)=1$;$T(n)=9T(\frac{n}{3})+n^3$ for $n>1.$

$($c$)T(1)=1$;$T(n)=T(\frac{n}{5})+T(3\frac{n}{4})+n$ for $n>1.$ Hint: Draw the recursion tree. Examine of the sum of costs across a level. Guess a solution. Then, try to prove by induction.