Certainly! Let's take a step-by-step approach to solve for the first term of the arithmetic sequence given the common difference [tex]\(d = -3\)[/tex] and the 8th term ([tex]\(a_8\)[/tex]) is 30.
### Step-by-Step Solution
1. Identify the parameters of the problem:
- Common difference ([tex]\(d\)[/tex]): [tex]\(-3\)[/tex]
- 8th term ([tex]\(a_8\)[/tex]): 30
- Term position ([tex]\(n\)[/tex]): 8
2. Recall the formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence:
[tex]\[
a_n = a + (n-1) \cdot d
\][/tex]
where:
- [tex]\(a_n\)[/tex] is the [tex]\(n\)[/tex]-th term
- [tex]\(a\)[/tex] is the first term
- [tex]\(d\)[/tex] is the common difference
- [tex]\(n\)[/tex] is the term position
3. Substitute the given values into the formula:
[tex]\[
30 = a + (8-1) \cdot (-3)
\][/tex]
Simplifying inside the parentheses:
[tex]\[
30 = a + 7 \cdot (-3)
\][/tex]
[tex]\[
30 = a - 21
\][/tex]
4. Solve for the first term ([tex]\(a\)[/tex]):
[tex]\[
a - 21 = 30
\][/tex]
Adding 21 to both sides to isolate [tex]\(a\)[/tex]:
[tex]\[
a = 30 + 21
\][/tex]
[tex]\[
a = 51
\][/tex]
### Conclusion
The first term of the arithmetic sequence is:
[tex]\[
a = 51
\][/tex]
So, given the common difference [tex]\(d = -3\)[/tex] and that the 8th term is 30, we find that the first term [tex]\(a\)[/tex] is 51.
