The expression provided in the image is asking for the sum $S_{18}$ of the first 18 terms of the arithmetic sequence $3.2 + 2.9 + 2.6 + 2.3 + \ldots$.

Steps to Solve:

1. Identify the first term ($a_1$) and common difference ($d$):

   • First term $a_1 = 3.2$
   • Common difference $d = 2.9 - 3.2 = -0.3$

2. Use the formula for the sum of the first $n$ terms of an arithmetic sequence: $S_n = \frac{n}{2} \times [2a_1 + (n-1)d]$ where:

   • $S_n$ is the sum of the first $n$ terms.
   • $n = 18$
   • $a_1 = 3.2$
   • $d = -0.3$

3. Substitute the values into the formula: $S_{18} = \frac{18}{2} \times [2(3.2) + (18-1)(-0.3)]$ $S_{18} = 9 \times [6.4 + 17(-0.3)]$ $S_{18} = 9 \times [6.4 - 5.1]$ $S_{18} = 9 \times 1.3 = 11.7$

So, the sum $S_{18}$ for this arithmetic sequence is 11.7.

Would you like further details or have any questions?

Related Questions:

1. How do you derive the formula for the sum of an arithmetic sequence?
2. What are some common properties of arithmetic sequences?
3. Can you explain how the common difference affects the sum of a sequence?
4. How would the sum change if the sequence had more or fewer terms?
5. What is the difference between an arithmetic and a geometric sequence?

Tip:

Always check the common difference in a sequence carefully—it dictates whether the sequence is increasing, decreasing, or constant.