1. **State the problem:** We have three consecutive positive integers with $n$ as the middle integer. Let the three integers be $(n-1)$, $n$, and $(n+1)$. We multiply these three integers and then add $n$ to the product. We want to prove that the resulting number is a perfect cube. 2. **Express the product:** The product of the three consecutive integers is: $$ (n - 1) \times n \times (n + 1) $$ This can be rewritten using the formula for the product of three consecutive integers centered at $n$: $$ (n - 1) n (n + 1) = n (n^2 - 1) $$ 3. **Add $n$ to the product:** The total expression is: $$ n (n^2 - 1) + n = n (n^2 - 1 + 1) = n (n^2) = n^3 $$ 4. **Interpret the result:** We simplified the expression to $n^3$, which is by definition a perfect cube. **Final answer:** The result of multiplying the three consecutive integers and then adding the middle integer is always $n^3$, a perfect cube.