To solve this problem and describe the given geometric sequence, let’s work through the steps required for constructing the recursively defined function.
The given sequence is:
[tex]\[
-21, 63, -189, 567, \ldots
\][/tex]
First, let's establish a few key elements of the sequence:
1. First term of the sequence ([tex]\( f(1) \)[/tex]):
The first term is [tex]\(-21\)[/tex].
2. Common ratio ([tex]\( r \)[/tex]):
To determine the common ratio, divide the second term by the first term.
[tex]\[
r = \frac{63}{-21} = -3
\][/tex]
Now, we have:
- The first term, [tex]\( f(1) = -21 \)[/tex]
- The common ratio, [tex]\( r = -3 \)[/tex]
Given a geometric sequence, the general formula for the [tex]\(n\)[/tex]-th term is:
[tex]\[
f(n) = f(1) \cdot r^{(n-1)}
\][/tex]
### Recursively Defined Function
A recursively defined function specifies each term of the sequence based on the previous term. For this geometric sequence, we can define it as follows:
- The first term is provided directly:
[tex]\[
f(1) = -21
\][/tex]
- From the second term onward, each term is the product of the previous term and the common ratio:
[tex]\[
f(n) = f(n-1) \cdot (-3) \quad \text{for } n \geq 2
\][/tex]
Combining these, the recursively defined function for the sequence is:
[tex]\[
\begin{cases}
f(1) = -21 \\
f(n) = f(n-1) \cdot (-3) & \text{for } n \geq 2
\end{cases}
\][/tex]
This completes the recursive definition of the given geometric sequence.
