To find the explicit formula for the sequence given by the recursive formula:
[tex]\[
f(n)=\left\{\begin{array}{l}
f(1)=30 \\
f(n)=f(n-1)-8 \text{ if } n>1
\end{array}\right.
\][/tex]
let’s work through the steps of converting this recursive formula into an explicit formula.
### Step-by-Step Solution
1. Understand the Recursive Formula:
The recursive formula tells us two things:
- The initial term [tex]\( f(1) = 30 \)[/tex]
- Each subsequent term is obtained by subtracting 8 from the previous term, i.e., [tex]\( f(n) = f(n-1) - 8 \)[/tex] for [tex]\( n > 1 \)[/tex].
2. Identify the Pattern:
By repeatedly applying the recursive formula, we can identify a pattern:
- [tex]\( f(2) = f(1) - 8 = 30 - 8 = 22 \)[/tex]
- [tex]\( f(3) = f(2) - 8 = 22 - 8 = 14 \)[/tex]
- [tex]\( f(4) = f(3) - 8 = 14 - 8 = 6 \)[/tex]
It becomes clear that each term decreases by 8 compared to the previous term. This indicates that the sequence is arithmetic with a common difference of -8.
3. General Formula for Arithmetic Sequence:
The formula for the nth term of an arithmetic sequence is generally expressed as:
[tex]\[
f(n) = f(1) + (n - 1) \cdot d
\][/tex]
where [tex]\( f(1) \)[/tex] is the first term and [tex]\( d \)[/tex] is the common difference.
4. Substitute the Known Values:
- [tex]\( f(1) = 30 \)[/tex]
- [tex]\( d = -8 \)[/tex]
Now, substituting these values into the general formula:
[tex]\[
f(n) = 30 + (n - 1) \cdot (-8)
\][/tex]
5. Simplify the Expression:
Simplify the expression to obtain the explicit formula:
[tex]\[
f(n) = 30 + (n - 1) \cdot (-8)
\][/tex]
[tex]\[
f(n) = 30 - 8 \cdot (n - 1)
\][/tex]
### Conclusion
The explicit formula for the given sequence is:
[tex]\[
f(n) = 30 - 8 (n - 1)
\][/tex]
Therefore, the correct choice from the given options is:
[tex]\[
f(n) = 30 - 8 (n - 1)
\][/tex]
