We saw that the solution of $T(n) = T(\lceil n/2 \rceil) + n$ is $O(lg\ n)$. Show that the solution of this recurrence is also $\Omega(n\ lg\ n)$. Conclude that the solution is $\Theta(n\log \ n)$.

Using the master method, you can show that the solution to the recurrence $T(n) = 4T(n/2) +n^2$ is $T(n) = \Theta(n^2)$.Show that a substitution proof with the assumption $T(n) \leq cn^2$ fails. Then show how to subtract off a lower-order term to make a substitution proof work.

Using the master method, you can show that the solution to the recurrence $T(n) = 4T(n/3) +n$ is $T(n) = O(n^{log_34})$.Show that a substitution proof with the assumption $T(n) \leq cn^{log_34}$ fails. Then show how to subtract off a lower-order term to make a substitution proof work.