To find the sum of the first 15 terms of the arithmetic sequence given by 13, 8, 3, -2, ..., we can follow the steps below:
### Step 1: Identify the First Term and Common Difference
First, we identify the first term (denoted as [tex]\(a\)[/tex]) and the common difference (denoted as [tex]\(d\)[/tex]):
- The first term, [tex]\(a\)[/tex], is 13.
- The common difference, [tex]\(d\)[/tex], can be found by subtracting the first term from the second term:
[tex]\[ d = 8 - 13 = -5 \][/tex]
### Step 2: Determine the Number of Terms
The number of terms in our sequence is given as [tex]\(n = 15\)[/tex].
### Step 3: Use the Formula for the Sum of the First [tex]\(n\)[/tex] Terms
The formula to find the sum of the first [tex]\(n\)[/tex] terms of an arithmetic sequence is:
[tex]\[ S_n = \frac{n}{2} (2a + (n-1)d) \][/tex]
### Step 4: Substitute the Terms into the Formula
Substitute [tex]\(a = 13\)[/tex], [tex]\(d = -5\)[/tex], and [tex]\(n = 15\)[/tex] into the formula:
[tex]\[ S_{15} = \frac{15}{2} \left(2 \times 13 + (15-1) \times (-5)\right) \][/tex]
### Step 5: Calculate the Intermediate Values
First, compute the values inside the parentheses:
[tex]\[ 2 \times 13 = 26 \][/tex]
[tex]\[ 15 - 1 = 14 \][/tex]
[tex]\[ 14 \times (-5) = -70 \][/tex]
Now sum these values:
[tex]\[ 26 + (-70) = -44 \][/tex]
### Step 6: Compute the Final Sum
Now multiply by [tex]\(\frac{15}{2}\)[/tex]:
[tex]\[ S_{15} = \frac{15}{2} \times (-44) \][/tex]
[tex]\[ S_{15} = 15 \times -22 \][/tex]
[tex]\[ S_{15} = -330 \][/tex]
So, the sum of the first 15 terms of the arithmetic sequence is [tex]\(-330\)[/tex].
