Solving Recurrences Using Master Method
1) $T(n) =  3T(\frac{n}{2}) + n \in \Theta(n\ln(n))$

a = 3, b= 2 and f(n) = $\Theta(n)$

$n^{\log_b a} = n^{\log_2 3} &gt; n^{\log_2 2}  = n = f(n) \therefore f(n) \in \Theta(n^{\log_2 3})$

(Case 1) Proved False

2) It is not in the Master Theorem format $T(n) =  aT(\frac{n}{b}) + f(n) \therefore$

It is not possible to say if the statement is true or false using the Master Theorem

3)  $T(n) =  2T(\frac{n}{3}+1) + n  = 2T(\frac{n+3}{3}) + n \in \Theta(n\ln(n))$

a = 2, b= 3 and f(n) = $\Theta(n)$

$n^{\log_b a} = n^{\log_3 2} &lt; n^{\log_3 3}  = n = f(n) \therefore$

Case 3.

Now we could check if it is the case that $T(n) \in \Theta(f(n))$, but we don't need to, because the only way it would be $\Theta(n\log n)$ was if it was case 2.