To find the recursive rule for the given geometric sequence [tex]\( f(n) = 13 (1.3)^{n-1} \)[/tex], let's analyze each part of the sequence step-by-step.
1. Initial Term:
The initial term [tex]\( f(1) \)[/tex] is given by plugging in [tex]\( n = 1 \)[/tex] into the formula:
[tex]\[
f(1) = 13 \cdot (1.3)^{1-1} = 13 \cdot (1.3)^0 = 13 \cdot 1 = 13
\][/tex]
So, we have:
[tex]\[
f(1) = 13
\][/tex]
2. Recursive Relationship:
A geometric sequence has a common ratio between consecutive terms. For this sequence, the common ratio [tex]\( r \)[/tex] can be determined from the formula:
[tex]\[
f(n) = 13 \cdot (1.3)^{n-1}
\][/tex]
To find the relationship between [tex]\( f(n) \)[/tex] and [tex]\( f(n-1) \)[/tex], consider:
[tex]\[
f(n-1) = 13 \cdot (1.3)^{(n-1)-1} = 13 \cdot (1.3)^{n-2}
\][/tex]
To express [tex]\( f(n) \)[/tex] using [tex]\( f(n-1) \)[/tex]:
\begin{align}
f(n) &= 13 \cdot (1.3)^{n-1} \\
&= 13 \cdot (1.3)^{n-2} \cdot (1.3)^1 \\
&= (13 \cdot (1.3)^{n-2}) \cdot 1.3 \\
&= f(n-1) \cdot 1.3
\end{align}
Therefore, the recursive relationship is:
[tex]\[
f(n) = 1.3 \cdot f(n-1) \quad \text{for} \quad n \geq 2
\][/tex]
Combining both the initial term and the recursive relationship, the recursive rule for the geometric sequence is:
[tex]\[
f(1) = 13, \quad f(n) = 1.3 \cdot f(n-1) \quad \text{for} \quad n \geq 2
\][/tex]
Thus, the correct choice is:
\[
f(1)=13, \quad f(n)=1.3 \cdot f(n-1), \quad n\geq 2
\
