Thank you for pointing that out! Let's clarify the step where the expression \(5n\) comes from in problem 2.

We're given:
- \( S_n = 180 \)
- \( a_1 = 5 \)
- \( d = 5 \)

The sum of the first \( n \) terms of an arithmetic series is given by:

\[
S_n = \frac{n}{2} \times \left(2a_1 + (n-1)d\right)
\]

Substitute the given values:

\[
180 = \frac{n}{2} \times \left(2(5) + (n-1)(5)\right)
\]

Now, let's break this down properly:

1. First, compute the expression inside the parentheses:
   \[
   2(5) + (n-1)(5)
   \]
   \[
   = 10 + 5(n-1)
   \]

2. Distribute \(5\) across \((n-1)\):
   \[
   = 10 + 5n - 5
   \]

3. Simplify the expression:
   \[
   = 5n + 5
   \]

Thus, the equation becomes:

\[
180 = \frac{n}{2} \times (5n + 5)
\]

This is where the \(5n + 5\) comes from. We then proceed to solve the equation as shown previously.

Would you like to revisit any steps or have more details?

Here are 5 related questions to deepen your understanding:
1. How would the expression change if the common difference \( d \) was a different number?
2. What if the initial term \( a_1 \) was negative—how would that affect the sum?
3. Can you calculate the sum \( S_n \) if you know \( n \), \( a_1 \), and \( d \)?
4. How do you derive the formula for the sum \( S_n \) from the general definition of an arithmetic series?
5. What happens to the equation if both \( a_1 \) and \( d \) are equal?

**Tip:** When simplifying algebraic expressions, ensure each step logically follows from the previous one, especially when dealing with distribution and combining like terms.

Discover how 5n is derived in the calculation of the sum of an arithmetic series. This problem involves key concepts in arithmetic series and algebra, using the formula for the sum of arithmetic series. Suitable for students in Grades 9-12.

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