To solve this problem, we need to find the sum of the given arithmetic sequence. Let’s break it down:

The sequence provided:

$-5, -\frac{5}{2}, 0, \frac{5}{2}, 5, \dots, 50$

Step 1: Identify the first term ($a$) and common difference ($d$):

• The first term ($a$) is clearly $-5$.
• To find the common difference ($d$), subtract the first term from the second term: $d = -\frac{5}{2} - (-5) = -\frac{5}{2} + 5 = \frac{5}{2}.$ So, the common difference is $d = \frac{5}{2}$.

Step 2: Determine the total number of terms ($n$):

The last term of the sequence is $50$. Using the general formula for the $n$-th term of an arithmetic sequence: $a_n = a + (n - 1)d$ Substitute $a_n = 50$, $a = -5$, and $d = \frac{5}{2}$: $50 = -5 + (n - 1)\frac{5}{2}.$ Simplify: $50 + 5 = (n - 1)\frac{5}{2}.$ $55 = (n - 1)\frac{5}{2}.$ Multiply through by $2$ to eliminate the fraction: $110 = 5(n - 1).$ Divide by $5$: $n - 1 = 22.$ Add $1$ to both sides: $n = 23.$ So, the sequence has $23$ terms.

Step 3: Calculate the sum of the sequence ($S_n$):

The formula for the sum of an arithmetic sequence is: $S_n = \frac{n}{2}(a + a_n).$ Substitute $n = 23$, $a = -5$, and $a_n = 50$: $S_{23} = \frac{23}{2}(-5 + 50).$ Simplify: $S_{23} = \frac{23}{2}(45).$ $S_{23} = \frac{1035}{2}.$ $S_{23} = 517.5.$

Final Answer:

The sum of the sequence is: $\boxed{517.5}$

Would you like further details on any step?

Expanding questions:

1. What is the general formula for an arithmetic sequence, and how can it be applied here?
2. How is the common difference determined in an arithmetic sequence?
3. Why do we use the formula $S_n = \frac{n}{2}(a + a_n)$ to find the sum of an arithmetic sequence?
4. How do we verify the correctness of the total number of terms ($n$) in the sequence?
5. How does the sum formula differ for arithmetic sequences versus geometric sequences?

Tip:

Always double-check the arithmetic calculations, especially when fractions or large numbers are involved, to avoid errors in determining $n$ or the sum.