1. Stating the problem: Let the three consecutive positive integers be $n-1$, $n$, and $n+1$, where $n$ is the middle integer. 2. Writing the product of these three consecutive integers: $$ (n-1) \times n \times (n+1) $$ 3. We add $n$ to the product: $$ (n-1) \times n \times (n+1) + n $$ 4. Simplifying the product part first, note that: $$ (n-1)(n+1) = n^2 - 1 $$ So the product becomes: $$ n (n^2 - 1) = n^3 - n $$ 5. Now add $n$: $$ n^3 - n + n = n^3 $$ 6. The simplified expression is: $$ n^3 $$ 7. Since $n^3$ is a perfect cube (the cube of the integer $n$), the result is always a cube number. Thus, the expression $(n-1)n(n+1) + n$ simplifies and equals $n^3$, proving that the result is a cube number.