The given sequence is an arithmetic sequence:

$-9, 0, 9, 18, \dots$

We are tasked with finding $S_{20}$, the sum of the first 20 terms of this arithmetic sequence.

Step 1: Identify the first term and common difference

• The first term ($a_1$) is $-9$.
• The common difference ($d$) can be found by subtracting any term from its preceding term. For example: $d = 0 - (-9) = 9$

So, the common difference is $d = 9$.

Step 2: Use the formula for the sum of an arithmetic sequence

The formula for the sum of the first $n$ terms of an arithmetic sequence is: $S_n = \frac{n}{2} \times (2a_1 + (n-1)d)$ Where:

• $n = 20$
• $a_1 = -9$
• $d = 9$

Step 3: Substitute the values into the formula

$S_{20} = \frac{20}{2} \times \left( 2(-9) + (20-1)(9) \right)$ $S_{20} = 10 \times \left( -18 + 19(9) \right)$ $S_{20} = 10 \times \left( -18 + 171 \right)$ $S_{20} = 10 \times 153 = 1530$

Final Answer:

The sum of the first 20 terms of the sequence is $S_{20} = 1530$.

Would you like further clarification or additional details on the process? Here are some related questions you might find interesting:

1. How do you find the nth term of an arithmetic sequence?
2. What is the general formula for the sum of an arithmetic series?
3. Can arithmetic sequences have negative common differences?
4. How would the sum change if the common difference were halved?
5. What is the significance of the common difference in determining the growth of the sequence?

Tip: When dealing with arithmetic sequences, always check the common difference to verify it's consistent throughout the sequence.