To find the sum of the first 10 terms of the arithmetic sequence [tex]\( -8x, -5x, -2x, 1x, \ldots \)[/tex], we will follow these steps:
### Step 1: Identify the first term ([tex]\(a_1\)[/tex]) and common difference ([tex]\(d\)[/tex])
- The first term of the sequence is [tex]\(a_1 = -8x\)[/tex].
- The second term is [tex]\(-5x\)[/tex].
The common difference [tex]\(d\)[/tex] is calculated as follows:
[tex]\[ d = (-5x) - (-8x) = -5x + 8x = 3x \][/tex]
### Step 2: Determine the 10th term ([tex]\(a_{10}\)[/tex])
The [tex]\(n\)[/tex]-th term of an arithmetic sequence is given by:
[tex]\[ a_n = a_1 + (n-1) \cdot d \][/tex]
For the 10th term ([tex]\(n = 10\)[/tex]):
[tex]\[ a_{10} = -8x + (10-1) \cdot 3x \][/tex]
[tex]\[ a_{10} = -8x + 9 \cdot 3x \][/tex]
[tex]\[ a_{10} = -8x + 27x \][/tex]
[tex]\[ a_{10} = 19x \][/tex]
### Step 3: Calculate the sum of the first 10 terms ([tex]\(S_{10}\)[/tex])
The sum of the first [tex]\(n\)[/tex] terms of an arithmetic sequence is given by:
[tex]\[ S_n = \frac{n}{2} \cdot (a_1 + a_n) \][/tex]
For [tex]\(n = 10\)[/tex]:
[tex]\[ S_{10} = \frac{10}{2} \cdot (-8x + 19x) \][/tex]
[tex]\[ S_{10} = 5 \cdot (11x) \][/tex]
[tex]\[ S_{10} = 55x \][/tex]
### Conclusion
The sum of the first 10 terms of the arithmetic sequence [tex]\( -8x, -5x, -2x, 1x, \ldots \)[/tex] is [tex]\( 55x \)[/tex].
