Let's derive the explicit formula for the given arithmetic sequence step-by-step:
1. We are given that the first term [tex]\( a_1 \)[/tex] of the sequence is 8:
[tex]\[
a_1 = 8
\][/tex]
2. The sequence is defined recursively with the relationship:
[tex]\[
a_n = a_{n-1} - 2
\][/tex]
This indicates that the common difference [tex]\( d \)[/tex] is -2.
3. To find the explicit formula for the [tex]\( n \)[/tex]-th term of an arithmetic sequence, we use the general formula:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
4. Substituting the known values ([tex]\( a_1 = 8 \)[/tex] and [tex]\( d = -2 \)[/tex]) into the formula:
[tex]\[
a_n = 8 + (n - 1) \cdot (-2)
\][/tex]
5. Simplify the expression inside the parentheses:
[tex]\[
a_n = 8 - 2(n - 1)
\][/tex]
Therefore, the explicit formula for the given arithmetic sequence is:
[tex]\[
a_n = 8 - 2(n - 1)
\][/tex]
Among the provided choices, the correct formula is:
[tex]\[
a_n = 8 - 2(n - 1)
\][/tex]
So, the correct answer is:
[tex]\[
a_n = 8-2(n-1)
\][/tex]
