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Find the first term for each of the following:

a. [tex]d = -3, 8^{\text{th}} \text{ term} = 30[/tex]


Sagot :

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.
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