Question

Which recursive formula can be used to generate the sequence below, where f(1) = 3 and n ≥ 1?

3, –6, 12, –24, 48, …

f (n + 1) = –3 f(n )
f (n + 1) = 3 f(n )
f (n + 1) = –2 f(n )
f (n + 1) = 2 f(n)

Answer 1

[AI-generated answer — Bot GPT-4o]

To determine the correct recursive formula for the given sequence (3, –6, 12, –24, 48, …) where \( f(1) = 3 \) and \( n \geq 1 \), we can analyze how each term relates to the previous one.

1. Starting with the first term \( f(1) = 3 \).

2. The second term is \( -6 \). Notice that \(-6 = -2 \times 3\).

3. The third term is \( 12 \). Notice that \( 12 = -2 \times (-6)\).

4. The fourth term is \( -24 \). Notice that \( -24 = -2 \times 12\).

5. The fifth term is \( 48 \). Notice that \( 48 = -2 \times (-24)\).

From this pattern, it is clear that to get from one term to the next, you multiply by \(-2\).

Thus, the recursive formula which generates this sequence is:
\[ f(n+1) = -2 f(n) \]

So the correct option is:
\[ f(n+1) = -2 f(n) \]