https://file.aiquickdraw.com/mathbot/file/1725395281016uhbqcocx.jpg

The image shows a sequence for which you need to find \( S_{18} \), the sum of the first 18 terms. The sequence starts as follows: \( 3.2 + 2.9 + 2.6 + 2.3 + \dots \).

This sequence is an **arithmetic sequence** because the difference between consecutive terms is constant.

### Step 1: Identify the First Term and the Common Difference
- The first term (\( a_1 \)) is 3.2.
- The common difference (\( d \)) can be calculated as:
  \[
  d = 2.9 - 3.2 = -0.3
  \]

### Step 2: Use the Formula for the Sum of an Arithmetic Sequence
The sum \( S_n \) of the first \( n \) terms of an arithmetic sequence is given by:
\[
S_n = \frac{n}{2} \times [2a_1 + (n-1)d]
\]
where:
- \( n = 18 \)
- \( a_1 = 3.2 \)
- \( d = -0.3 \)

### Step 3: Substitute the Values
Substituting the values into the formula:
\[
S_{18} = \frac{18}{2} \times [2(3.2) + (18-1)(-0.3)]
\]

\[
S_{18} = 9 \times [6.4 + 17(-0.3)]
\]

\[
S_{18} = 9 \times [6.4 - 5.1]
\]

\[
S_{18} = 9 \times 1.3
\]

\[
S_{18} = 11.7
\]

### Final Answer
The sum of the first 18 terms, \( S_{18} \), is **11.7**.

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Would you like more details on this calculation, or do you have any other questions?

### Related Questions:
1. How would the sum change if the common difference were positive?
2. Can you find the sum \( S_{10} \) for the same sequence?
3. What is the general formula for the \( n \)-th term of this arithmetic sequence?
4. How do you determine whether a sequence is arithmetic or geometric?
5. What is the sum of the first 20 terms of this sequence?

### Tip:
Always check the sign of the common difference; it indicates whether the sequence is increasing or decreasing.

Find the sum S18 for the arithmetic sequence 3.2 + 2.9 + 2.6 + 2.3 + …

Find the sum of the first 18 terms in an arithmetic sequence starting with 3.2, 2.9, 2.6, and so on. This solution demonstrates the use of the arithmetic series sum formula, including step-by-step calculations. Suitable for students in Grades 9-10.

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