$($b$)$ Prove that the solution to the following recurrence is T $($n$)=\backslash$Theta $($n$).$ Assume T $(1)=1.$

T $($n$)=$ T $($n$/3)+$ T $(3$n$/5)+$ n$.$

$($Hint: One approach would be to ignore the floors and use a recursion tree. Another more formal approach for the upper bound would be to use induction. Set your hypothesis to be that T $($n$)<=$ cn for all n $<$ m$,$ for a constant c$.$ Then establish the induction step that T $($m$)<=$ cm and in the process derive a bound on the constant c$.$ Both approaches are acceptable.$)$