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Bot GPT-4o The sequence you provided is \( 3, -6, 12, -24, 48, \ldots \).
To find a recursive formula for this sequence, observe the relationship between consecutive terms.
1. The first term \( f(1) = 3 \).
2. The second term is \( -6 \), which is \( -2 \times 3 \).
3. The third term is \( 12 \), which is \( -2 \times -6 \).
4. The fourth term is \( -24 \), which is \( -2 \times 12 \).
Notice the pattern: each term is \( -2 \) times the previous term.
Therefore, the recursive formula for the sequence can be defined as:
\[ f(n) = -2 \times f(n-1) \quad \text{for} \quad n \geq 2 \]
And explicitly stating the initial condition:
\[ f(1) = 3. \]
Combining these, the recursive formula is:
\[
\begin{cases}
f(1) = 3 \\
f(n) = -2 \times f(n-1) \quad \text{for} \quad n \geq 2
\end{cases}
\]