To calculate the sum of the first 10 terms of the arithmetic sequence [tex]\(-8x, -5x, -2x, 1x, \ldots\)[/tex], follow these steps:
1. Identify the first term [tex]\(a\)[/tex]:
The first term [tex]\(a\)[/tex] of the sequence is [tex]\(-8x\)[/tex].
2. Determine the common difference [tex]\(d\)[/tex]:
The common difference [tex]\(d\)[/tex] can be found by subtracting the first term from the second term:
[tex]\[
d = (-5x) - (-8x) = -5x + 8x = 3x
\][/tex]
3. Specify the number of terms [tex]\(n\)[/tex]:
The problem asks for the sum of the first 10 terms, so [tex]\(n = 10\)[/tex].
4. Use the formula for the sum of the first [tex]\(n\)[/tex] terms of an arithmetic sequence:
The formula for the sum [tex]\(S_n\)[/tex] of the first [tex]\(n\)[/tex] terms of an arithmetic sequence is given by:
[tex]\[
S_n = \frac{n}{2} \left(2a + (n-1)d\right)
\][/tex]
Here, [tex]\(a = -8x\)[/tex], [tex]\(d = 3x\)[/tex], and [tex]\(n = 10\)[/tex].
5. Substitute the values into the formula:
[tex]\[
S_{10} = \frac{10}{2} \left(2(-8x) + (10-1)(3x)\right)
\][/tex]
6. Calculate each component step-by-step:
- Calculate [tex]\(2(-8x)\)[/tex]:
[tex]\[
2(-8x) = -16x
\][/tex]
- Calculate [tex]\((10-1)(3x)\)[/tex]:
[tex]\[
(10-1)(3x) = 9(3x) = 27x
\][/tex]
- Add the results:
[tex]\[
-16x + 27x = 11x
\][/tex]
7. Finalize the sum calculation:
- Use the resultant values in the final calculation for [tex]\(S_{10}\)[/tex]:
[tex]\[
S_{10} = \frac{10}{2}(11x) = 5(11x) = 55x
\][/tex]
Therefore, the sum of the first 10 terms of the arithmetic sequence [tex]\(-8x, -5x, -2x, 1x, \ldots\)[/tex] is [tex]\(\boxed{55x}\)[/tex].
