Certainly! Let's define the recursively defined function to describe the given geometric sequence:
The sequence provided is:
[tex]\[
-21, 63, -189, 567, \ldots
\][/tex]
### Determining the first term ([tex]\(a_1\)[/tex]):
The first term of the sequence ([tex]\(a_1\)[/tex]) is [tex]\( -21 \)[/tex].
### Identifying the common ratio:
To determine the common ratio, we divide the second term by the first term, the third term by the second term, and so on to confirm consistency.
[tex]\[
\frac{63}{-21} = -3
\][/tex]
[tex]\[
\frac{-189}{63} = -3
\][/tex]
[tex]\[
\frac{567}{-189} = -3
\][/tex]
We see that the common ratio is [tex]\( -3 \)[/tex].
### Formulating the recursive function:
We can now complete the recursive function definition for this sequence. The recursive definition has two parts: the base case and the recursive case.
#### Base Case:
[tex]\( f(1) \)[/tex] is the first term of the sequence.
[tex]\[
f(1) = -21
\][/tex]
#### Recursive Case:
For [tex]\( n \geq 2 \)[/tex], each term is obtained by multiplying the previous term [tex]\( f(n-1) \)[/tex] by the common ratio [tex]\( -3 \)[/tex].
[tex]\[
f(n) = -3 \cdot f(n-1) \quad \text{for} \; n = 2, 3, 4, \ldots
\][/tex]
### Final Formulation of the Recursive Function:
[tex]\[
f(1) = -21
\][/tex]
[tex]\[
f(n) = -3 \cdot f(n-1) \quad \text{for} \; n = 2, 3, 4, \ldots
\][/tex]
### Completion:
Therefore, the complete recursively defined function for the geometric sequence is:
[tex]\[
f(1) = -21
\][/tex]
[tex]\[
f(n) = -3 \cdot f(n-1) \quad \text{for} \; n = 2, 3, 4, \ldots
\][/tex]
