The given problem involves an arithmetic series with the terms $2, 5, 8, 11, \dots$, and the sum of the series is 60,000. The task is to find the number of terms in the series.

Step-by-step solution:

1. Identifying parameters of the arithmetic series:

   • First term ($a$) = 2
   • Common difference ($d$) = 5 - 2 = 3
   • Sum of the series ($S_n$) = 60,000

2. Formula for the sum of an arithmetic series: The sum of the first $n$ terms of an arithmetic series is given by: $S_n = \frac{n}{2} \left(2a + (n - 1)d\right)$ Plugging in the known values: $60,000 = \frac{n}{2} \left(2(2) + (n - 1)(3)\right)$ Simplifying the equation: $60,000 = \frac{n}{2} \left(4 + 3n - 3\right)$ $60,000 = \frac{n}{2} (3n + 1)$ Multiplying both sides by 2 to eliminate the fraction: $120,000 = n(3n + 1)$ Expanding the equation: $120,000 = 3n^2 + n$ Rearranging the equation: $3n^2 + n - 120,000 = 0$

3. Solving the quadratic equation: Use the quadratic formula: $n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 3$, $b = 1$, and $c = -120,000$. Substituting these values into the quadratic formula: $n = \frac{-1 \pm \sqrt{1^2 - 4(3)(-120,000)}}{2(3)}$ $n = \frac{-1 \pm \sqrt{1 + 1,440,000}}{6}$ $n = \frac{-1 \pm \sqrt{1,440,001}}{6}$ $n = \frac{-1 \pm 1200}{6}$ Taking the positive root (since $n$ must be positive): $n = \frac{-1 + 1200}{6} = \frac{1199}{6} = 199.83$ Rounding to the nearest whole number, $n = 200$.

Conclusion:

The number of terms in the series is $\boxed{200}$.

Let me know if you want further clarification or details. Here are 5 related questions to deepen your understanding:

1. What is the formula for the $n$-th term of an arithmetic series?
2. How do you find the sum of an arithmetic series if the last term is known?
3. Can you solve a similar problem for the sum of the series $3 + 7 + 11 + 15 + \dots$?
4. How does changing the common difference affect the number of terms needed to reach a certain sum?
5. What would happen to the sum of the series if the first term was doubled?

Tip: Always remember to check if the quadratic formula gives two possible solutions and choose the one that makes sense in the context of the problem.