$($a$)$ Let $n=2^k.$ Derive the exact solution of the following recurrence using the

repeated substitution method and by showing all steps:

$T(1)=1$

$T(n)=2T(\frac{n}{2})+nlgn,$ for $n>1$

$($b$)$ Which case of the Master Theorem can you apply to give an asymptotic

bound for $T(n)?$ Explain why.

$($c$)$ For the above recurrence, draw the recursion tree with proper labels. If possible solve it on pen and paper and upload it