1. **Problem statement:** We have two snails, Flash and Trovão, racing. Flash moves at a constant speed of 5 cm per minute. Trovão moves 1 cm in the first minute, then increases his distance by 1 cm each subsequent minute (2 cm in the second minute, 3 cm in the third, and so on). We want to find after how many minutes Trovão catches up to Flash. 2. **Formulas and rules:** - Distance covered by Flash after $n$ minutes: $$D_F = 5n$$ - Distance covered by Trovão after $n$ minutes is the sum of the first $n$ natural numbers: $$D_T = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}$$ 3. **Set the distances equal to find when Trovão catches Flash:** $$5n = \frac{n(n+1)}{2}$$ Multiply both sides by 2 to clear the denominator: $$2 \times 5n = n(n+1)$$ $$10n = n^2 + n$$ 4. **Bring all terms to one side:** $$n^2 + n - 10n = 0$$ $$n^2 - 9n = 0$$ 5. **Factor the equation:** $$n(n - 9) = 0$$ 6. **Solve for $n$:** $$n = 0 \quad \text{or} \quad n = 9$$ Since $n=0$ means the start, the meaningful solution is $n=9$. 7. **Interpretation:** After 9 minutes, Trovão catches up to Flash. **Final answer:** $$\boxed{9}$$ minutes.