Please solve the following $2$ problems. $1.$ Using $\backslash$Theta $-$notation, provide asymptotically tight bounds in terms of n for the solution to each of the following recurrences. Assume each recurrence T$($n$)$ has a non$-$trivial base case of T$($n$)=\backslash$Theta $(1)$ for all n $<$ n$0$ where n$0$ is a suitably large constant. For example, if asked to solve T$($n$)=2$T$($n$/2)+$ n$,$ then your answer should be $\backslash$Theta $($n log n$).$ Give a brief explanation for each solution.

$($a$)$ A$($n$)=4$A$($n$/2)+$ n $2$

$($b$)$ B$($n$)=7$B$($n$/2)+$ n $3$

$($c$)$ C$($n$)=5$C$($n$/3)+$ n

$($d$)$ D$($n$)=$ D$($n$/4)+$ D$($n$/2)+$ n

$($e$)$ E$($n$)=3$E$($n$/3)+$ n lg n