Question
write a recursive rule for the sequence: f(n)=3x(10)^n^-1
Answers
GPT-4o mini
It seems like there might be a slight misunderstanding in the notation of your function \( f(n) \). To clarify and create a recursive rule, I will assume that you meant to define a sequence based on a formula involving powers of 10.
From your notation \( f(n) = 3 \times (10^{n-1}) \), the function represents a sequence where each term is related to a power of 10.
The explicit rule you've given can be simplified as:
\[
f(n) = 3 \cdot 10^{n-1}
\]
Now let's derive a recursive rule from this explicit formula. To find the recursive definition, we need to express \( f(n) \) in terms of \( f(n-1) \).
Notice that:
\[
f(n-1) = 3 \cdot 10^{(n-1)-1} = 3 \cdot 10^{n-2}
\]
Now, to express \( f(n) \) in terms of \( f(n-1) \):
\[
f(n) = 10 \cdot f(n-1)
\]
So we have:
\[
f(n) = 10 \cdot f(n-1)
\]
Thus, the recursive rule can be summarized as:
\[
\begin{align*}
f(1) &= 3 \quad \text{(base case)} \\
f(n) &= 10 \cdot f(n-1) \quad \text{for } n > 1
\end{align*}
\]
This gives us the base case and the recursive relationship for the sequence.
From your notation \( f(n) = 3 \times (10^{n-1}) \), the function represents a sequence where each term is related to a power of 10.
The explicit rule you've given can be simplified as:
\[
f(n) = 3 \cdot 10^{n-1}
\]
Now let's derive a recursive rule from this explicit formula. To find the recursive definition, we need to express \( f(n) \) in terms of \( f(n-1) \).
Notice that:
\[
f(n-1) = 3 \cdot 10^{(n-1)-1} = 3 \cdot 10^{n-2}
\]
Now, to express \( f(n) \) in terms of \( f(n-1) \):
\[
f(n) = 10 \cdot f(n-1)
\]
So we have:
\[
f(n) = 10 \cdot f(n-1)
\]
Thus, the recursive rule can be summarized as:
\[
\begin{align*}
f(1) &= 3 \quad \text{(base case)} \\
f(n) &= 10 \cdot f(n-1) \quad \text{for } n > 1
\end{align*}
\]
This gives us the base case and the recursive relationship for the sequence.