To solve the problem of finding the arithmetic means in the given sequence [tex]\( -8, \_, \_, 0 \)[/tex], we'll follow several steps. Let's break it down step-by-step:
### Step 1: Identify the terms and properties
We are given a sequence with four terms where the first term ([tex]\(a_1\)[/tex]) is [tex]\(-8\)[/tex] and the last term ([tex]\(a_4\)[/tex]) is [tex]\(0\)[/tex]. There are missing second ([tex]\(a_2\)[/tex]) and third ([tex]\(a_3\)[/tex]) terms that we need to find.
### Step 2: Recognize it as an arithmetic sequence
An arithmetic sequence is one in which the difference between consecutive terms, known as the common difference ([tex]\(d\)[/tex]), is constant.
### Step 3: Determine the common difference
To find the common difference ([tex]\(d\)[/tex]), we use the formula that relates the terms of an arithmetic sequence:
[tex]\[ d = \frac{a_4 - a_1}{3} \][/tex]
Substituting the given terms:
[tex]\[ a_1 = -8 \][/tex]
[tex]\[ a_4 = 0 \][/tex]
[tex]\[ \text{number of terms} = 4 \][/tex]
### Step 4: Calculate the common difference
[tex]\[ d = \frac{0 - (-8)}{3} = \frac{8}{3} = 2.6666666666666665 \][/tex]
### Step 5: Find the missing terms
Now we use the common difference to find the missing terms.
Finding [tex]\(a_2\)[/tex]:
[tex]\[ a_2 = a_1 + d \][/tex]
[tex]\[ a_2 = -8 + 2.6666666666666665 \][/tex]
[tex]\[ a_2 = -5.333333333333334 \][/tex]
Finding [tex]\(a_3\)[/tex]:
[tex]\[ a_3 = a_2 + d \][/tex]
[tex]\[ a_3 = -5.333333333333334 + 2.6666666666666665 \][/tex]
[tex]\[ a_3 = -2.6666666666666674 \][/tex]
### Step 6: Verify the sequence
Finally, let's verify the complete sequence with the terms we found:
[tex]\[ -8, -5.333333333333334, -2.6666666666666674, 0 \][/tex]
This forms a valid arithmetic sequence with a common difference of approximately [tex]\(2.6666666666666665\)[/tex].
### Conclusion
The missing terms in the sequence are:
[tex]\[ -5.333333333333334 \text{ and } -2.6666666666666674 \][/tex]
Thus, the complete sequence is:
[tex]\[ -8, -5.333333333333334, -2.6666666666666674, 0 \][/tex]
