1. **State the problem:** Solve the equation $\frac{n(n+1)}{2} = 210$ for $n$. 2. **Formula and explanation:** This equation represents the formula for the sum of the first $n$ natural numbers: $$S = \frac{n(n+1)}{2}$$ where $S$ is the sum. We need to find $n$ such that the sum equals 210. 3. **Multiply both sides by 2 to eliminate the denominator:** $$\cancel{2} \times \frac{n(n+1)}{\cancel{2}} = 210 \times 2$$ $$n(n+1) = 420$$ 4. **Rewrite as a quadratic equation:** $$n^2 + n - 420 = 0$$ 5. **Solve the quadratic equation using the quadratic formula:** The quadratic formula is $$n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=1$, and $c=-420$. 6. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 1^2 - 4 \times 1 \times (-420) = 1 + 1680 = 1681$$ 7. **Find the square root of the discriminant:** $$\sqrt{1681} = 41$$ 8. **Calculate the two possible values for $n$:** $$n = \frac{-1 \pm 41}{2}$$ 9. **Evaluate each case:** - For the plus sign: $$n = \frac{-1 + 41}{2} = \frac{40}{2} = 20$$ - For the minus sign: $$n = \frac{-1 - 41}{2} = \frac{-42}{2} = -21$$ 10. **Interpret the solutions:** Since $n$ represents a count of natural numbers, it must be positive. Therefore, the valid solution is: $$\boxed{20}$$