Determining Asymptotic Tight Bound
If the number of steps executed in the $i$th iteration is $i^2$, then the number of steps overall will be asymptotic to
$$
\sum_{i=1}^n (C+i^2) = \Theta(n^3),
$$
for some constant $C$ that comes from the loop control (i.e., incrementing $i$ and checking whether it equals $n$).

To see that $\sum_{i=1}^n (C+i^2) = \Theta(n^3)$, use
$$
\sum_{i=1}^n (C+i^2) \leq n(C+n^2) = Cn+n^3 = O(n^3), \\
\sum_{i=1}^n (C+i^2) \geq \sum_{i=n/2}^n i^2 \geq (n/2) (n/2)^2 = n^3/8 = \Omega(n^3).
$$
(In fact, $\sum_{i=1}^n i^2 \approx n^3/3$, as can be seen by approximating it by the integral $\int_1^n i^2 \, di$, or by using the exact formula $\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$.)