1. **State the problem:** We have an arithmetic sequence $2, 4, 6, 8, \ldots$ and want to find how many terms sum to 600. 2. **Recall the formula for the sum of an arithmetic sequence:** $$S_n = \frac{n}{2} (a_1 + a_n)$$ where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, and $a_n$ is the $n$th term. 3. **Identify the sequence parameters:** - First term $a_1 = 2$ - Common difference $d = 2$ - $n$th term formula: $a_n = a_1 + (n-1)d = 2 + 2(n-1) = 2n$ 4. **Express the sum in terms of $n$:** $$S_n = \frac{n}{2} (a_1 + a_n) = \frac{n}{2} (2 + 2n) = \frac{n}{2} \times 2(1 + n) = n(n+1)$$ 5. **Set the sum equal to 600 and solve for $n$:** $$n(n+1) = 600$$ $$n^2 + n - 600 = 0$$ 6. **Solve the quadratic equation:** Using the quadratic formula $n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=1$, $b=1$, $c=-600$: $$n = \frac{-1 \pm \sqrt{1^2 - 4 \times 1 \times (-600)}}{2} = \frac{-1 \pm \sqrt{1 + 2400}}{2} = \frac{-1 \pm \sqrt{2401}}{2}$$ $$\sqrt{2401} = 49$$ So, $$n = \frac{-1 \pm 49}{2}$$ 7. **Find the positive solution:** $$n = \frac{-1 + 49}{2} = \frac{48}{2} = 24$$ 8. **Interpretation:** The number of terms needed to sum to 600 is $\boxed{24}$.