The given arithmetic sequence is $2, 4, 6, 8, \ldots$. To find how many terms sum up to 600, we can use the formula for the sum of the first $n$ terms of an arithmetic sequence:

$S_n = \frac{n}{2} \times (a_1 + a_n)$

where:

• $S_n$ is the sum of the first $n$ terms,
• $n$ is the number of terms,
• $a_1$ is the first term,
• $a_n$ is the $n$-th term.

We know:

• $S_n = 600$,
• $a_1 = 2$,
• The common difference $d = 2$ (since the difference between consecutive terms is 2).

The $n$-th term $a_n$ of an arithmetic sequence can be expressed as:

$a_n = a_1 + (n-1) \times d$ Substituting the known values: $a_n = 2 + (n-1) \times 2 = 2n$

Now, substitute the sum formula $S_n = 600$ and the expressions for $a_1$ and $a_n$:

$600 = \frac{n}{2} \times (2 + 2n)$

Simplify the equation:

$600 = \frac{n}{2} \times (2n + 2)$ $600 = \frac{n}{2} \times 2(n + 1)$ $600 = n(n + 1)$

Now solve for $n$:

$n(n + 1) = 600$ $n^2 + n - 600 = 0$

Solve the quadratic equation using the quadratic formula:

$n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ For the equation $n^2 + n - 600 = 0$, $a = 1$, $b = 1$, and $c = -600$:

$n = \frac{-1 \pm \sqrt{1^2 - 4 \times 1 \times (-600)}}{2 \times 1}$ $n = \frac{-1 \pm \sqrt{1 + 2400}}{2}$ $n = \frac{-1 \pm \sqrt{2401}}{2}$ $n = \frac{-1 \pm 49}{2}$

This gives two solutions:

$n = \frac{-1 + 49}{2} = \frac{48}{2} = 24$ $n = \frac{-1 - 49}{2} = \frac{-50}{2} = -25 \quad (\text{not a valid solution since } n \geq 0)$

Thus, $n = 24$. Therefore, 24 terms of the sequence sum up to 600.

Would you like a more detailed explanation or have any other questions? Here are some related questions:

1. How do you calculate the sum of an arithmetic sequence?
2. What happens if the common difference in an arithmetic sequence is negative?
3. How do you find the general term of an arithmetic sequence?
4. What is the relationship between the number of terms and the sum in a geometric sequence?
5. How can the formula for the sum of an arithmetic series be derived?

Tip: Always check whether both solutions from the quadratic formula are physically meaningful. Negative or non-integer values of $n$ are not valid for counting terms in a sequence!