1. **State the problem:** We need to find the value of $n$ in a triangle with angles $(n + 17)^\circ$, $31^\circ$, and $75^\circ$. 2. **Recall the triangle angle sum rule:** The sum of the interior angles of a triangle is always $180^\circ$. 3. **Set up the equation:** $$ (n + 17) + 31 + 75 = 180 $$ 4. **Simplify the constants:** $$ n + 17 + 31 + 75 = 180 $$ $$ n + 123 = 180 $$ 5. **Isolate $n$ by subtracting 123 from both sides:** $$ n + \cancel{123} - \cancel{123} = 180 - 123 $$ $$ n = 57 $$ 6. **Final answer:** $$ \boxed{57} $$