1. **State the problem:** We are given the formula for the sum $S_n = n^2 - 41n$ and need to show that $S_n = 0$ for some value(s) of $n$. 2. **Set the sum equal to zero:** To find when $S_n = 0$, solve the equation: $$n^2 - 41n = 0$$ 3. **Factor the equation:** Factor out $n$: $$n(n - 41) = 0$$ 4. **Solve for $n$:** Set each factor equal to zero: - $n = 0$ - $n - 41 = 0 \Rightarrow n = 41$ 5. **Interpretation:** The sum $S_n$ equals zero when $n = 0$ or $n = 41$. Since $n$ typically represents a positive integer count, the meaningful solution is $n = 41$. **Final answer:** $$S_n = 0 \text{ when } n = 0 \text{ or } n = 41$$ Thus, $S_n = 0$ at $n=41$ for the given formula.