Consider the equality $\sum_{i=0}^{n} i^3 = X$ and the following choices for $X$ I. $\Theta(n^4)$ II. $\Theta(n^5)$ III. $O(n^5)$ IV. $\Omega(n^3)$ The equality above remains correct if $X$ is replaced by

The sum of cubes of the first $n$ integers is given by the formula:

The leading term is $\frac{n^4}{4}$, so the exact growth rate is $\Theta(n^4)$.

Let us analyze the given asymptotic choices for $X$:

• I. $\Theta(n^4)$: True, since $\frac{n^4 + 2n^3 + n^2}{4} = \Theta(n^4)$.

• II. $\Theta(n^5)$: False, because the polynomial has degree $4$, not $5$.

• III. $O(n^5)$: True, because $O(n^5)$ represents an upper bound and $n^4 \le c \cdot n^5$ for all sufficiently large $n$.

• IV. $\Omega(n^3)$: True, because $\Omega(n^3)$ represents a lower bound and $n^4 \ge c \cdot n^3$ for all sufficiently large $n$.

Therefore, replacing $X$ with I, III, or IV keeps the equality correct, but II does not [1]. This corresponds to option (C) [1].