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.

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.