Recurrence Equation upper limit problem
Yes, you could solve the summation in exact formula. Yes, you could obtain $T(n)=\Theta(n^3)$, a tighter asymptotic bound. Your professor would and should not forbid you to do  that.

The message from your professor could be that an upper bound such as $O(n^3)$ is as good as $\Theta(n^3)$ more often than not. For example, once we know an algorithm runs in $O(n^3)$ time without a huge constant multiplier on a problem of input size $n$ less than 1000, then we can be comfortable starting implementing the algorithm,  without worrying much that the code may take hours to run. A tighter asymptotic bound such as  $\Theta(n^3)$ will not help us move ahead significantly further. On the other hand, such a tighter asymptotic usually comes with a heavier price, which is not much heavier in this particular problem, though.