To find the explicit formula for the given arithmetic sequence, let's analyze the problem step by step.
We are given the recursive formula:
[tex]\[
\left\{\begin{array}{l}
a_1=4 \\
a_n=a_{n-1}-7
\end{array}\right.
\][/tex]
Step 1: Identify the first term ([tex]\(a_1\)[/tex]).
The first term [tex]\(a_1\)[/tex] is given as [tex]\(4\)[/tex].
Step 2: Identify the common difference ([tex]\(d\)[/tex]).
The recursive formula [tex]\(a_n = a_{n-1} - 7\)[/tex] indicates that each term is obtained by subtracting 7 from the previous term. Therefore, the common difference [tex]\(d\)[/tex] is [tex]\(-7\)[/tex].
Step 3: Write the general formula for an arithmetic sequence.
The general form of an arithmetic sequence is given by:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
Step 4: Substitute the known values into the general formula.
We have [tex]\(a_1 = 4\)[/tex] and [tex]\(d = -7\)[/tex]. Plug these values into the general formula:
[tex]\[
a_n = 4 + (n - 1) \cdot (-7)
\][/tex]
Step 5: Simplify the formula.
Now, we simplify the expression:
[tex]\[
a_n = 4 + (n - 1) \cdot (-7)
\][/tex]
Distribute [tex]\(-7\)[/tex] within the parentheses:
[tex]\[
a_n = 4 + n \cdot (-7) - (-7)
\][/tex]
[tex]\[
a_n = 4 - 7n + 7
\][/tex]
Combine like terms:
[tex]\[
a_n = 11 - 7n
\][/tex]
Hence, the explicit formula for the arithmetic sequence is:
[tex]\[
a_n = 11 - 7n
\][/tex]
Step 6: Verify the correct option.
Among the given options:
- A. [tex]\(a_n=4+(n-7)(-1)\)[/tex]
- B. [tex]\(a_n=(-7)+(n-1) 4\)[/tex]
- C. [tex]\(a_n=4+(n-1)(-7)\)[/tex]
- D. [tex]\(a_n=(-1)+(n-4)(-7)\)[/tex]
The correct explicit formula [tex]\(\boxed{C}\)[/tex], which matches our derived formula:
[tex]\[
a_n = 4 + (n - 1)(-7)
\][/tex]
Thus, the explicit formula for the given arithmetic sequence is [tex]\(\boxed{C}\)[/tex].
