An arithmetic series is the sum of the terms in an arithmetic sequence, where the difference between consecutive terms is constant.

In the given sequence: $-17, -12, -7, \dots, 33$, let's break down the problem:

Step 1: Identify the common difference ($d$)

The common difference $d$ is calculated as: $d = -12 - (-17) = 5$

Step 2: Find the first term ($a$)

The first term $a$ is: $a = -17$

Step 3: Determine the number of terms ($n$)

The $n$-th term of an arithmetic sequence can be calculated using the formula: $a_n = a + (n-1)d$ Given $a_n = 33$, we can solve for $n$: $33 = -17 + (n-1)(5)$ $33 + 17 = 5(n-1)$ $50 = 5(n-1)$ $n-1 = 10 \implies n = 11$ So, there are 11 terms in the sequence.

Step 4: Write the arithmetic series in summation notation

The sum of the first $n$ terms ($S_n$) of an arithmetic series can be expressed as: $S_n = \sum_{k=1}^{n} \left( a + (k-1)d \right)$ For the given series: $S_{11} = \sum_{k=1}^{11} \left( -17 + (k-1)5 \right)$

Conclusion

The arithmetic series for the sequence $-17, -12, -7, \dots, 33$ in summation notation is: $\sum_{k=1}^{11} \left( -17 + (k-1)5 \right)$

Would you like further details on arithmetic series or have any questions?

Here are 5 related questions to explore further:

1. How do you calculate the sum of an arithmetic series?
2. What is the difference between an arithmetic series and an arithmetic sequence?
3. Can you derive the formula for the sum of an arithmetic series?
4. How would the series change if the common difference were negative?
5. What are some real-world applications of arithmetic series?

Tip: Always verify the number of terms in the sequence before writing it in summation notation.