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.