Let's solve the problem by writing the Arithmetic Sequence using an explicit formula given the provided options.
First, we need to recognize that for any arithmetic sequence, each term [tex]\(a_n\)[/tex] can be expressed based on the first term [tex]\(a_1\)[/tex] and the common difference [tex]\(d\)[/tex]. The formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence is:
[tex]\[ a_n = a_1 + (n - 1)d \][/tex]
From the problem, we are given:
- The first term [tex]\(a_1 = 12\)[/tex]
- The common difference [tex]\(d = -3\)[/tex]
Using the arithmetic sequence formula:
[tex]\[ a_n = a_1 + (n - 1)d \][/tex]
Now substituting [tex]\(a_1 = 12\)[/tex] and [tex]\(d = -3\)[/tex] into the formula:
[tex]\[ a_n = 12 + (n - 1)(-3) \][/tex]
[tex]\[ a_n = 12 - 3(n - 1) \][/tex]
So, the correct formula to use is:
[tex]\[ a_n = 12 - 3(n - 1) \][/tex]
Now let's verify the given options with this formula:
1. [tex]\( a_n = 12 - 3(n - 1) \)[/tex]
- This matches our derived formula.
2. [tex]\( a_n = 3 + 12(n - 1) \)[/tex]
- This does not match the derived formula.
3. [tex]\( a_n = 12 + 3(n - 1) \)[/tex]
- This does not match the derived formula.
4. [tex]\( a_n = 3 - 12(n - 1) \)[/tex]
- This does not match the derived formula.
Therefore, the correct explicit formula for the given arithmetic sequence is:
[tex]\[ a_n = 12 - 3(n - 1) \][/tex]
