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Which recursive formula can be used to generate the sequence below, where [tex]f(1) = 3[/tex] and [tex]n \geq 1[/tex]?

[tex]\[ 3, -6, 12, -24, 48, \ldots \][/tex]

A. [tex]f(n+1) = -3f(n)[/tex]

B. [tex]f(n+1) = 3f(n)[/tex]

C. [tex]f(n+1) = -2f(n)[/tex]

D. [tex]f(n+1) = 2f(n)[/tex]

Sagot :

GinnyAnsweridn
Let's analyze the given sequence: [tex]$3, -6, 12, -24, 48, \ldots$[/tex]

Identify [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 3 \][/tex]

Now, let's observe the pattern in the sequence by calculating the subsequent terms.

1. To calculate [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = -6 \][/tex]

2. To calculate [tex]\( f(3) \)[/tex]:
[tex]\[ f(3) = 12 \][/tex]

3. To calculate [tex]\( f(4) \)[/tex]:
[tex]\[ f(4) = -24 \][/tex]

4. To calculate [tex]\( f(5) \)[/tex]:
[tex]\[ f(5) = 48 \][/tex]

We can look at these calculations to find the relationship between successive terms:

[tex]\[ f(2) = -6 = -3 \times 3 = -3 \times f(1) \][/tex]
[tex]\[ f(3) = 12 = -3 \times -6 = -3 \times f(2) \][/tex]
[tex]\[ f(4) = -24 = -3 \times 12 = -3 \times f(3) \][/tex]
[tex]\[ f(5) = 48 = -3 \times -24 = -3 \times f(4) \][/tex]

From these calculations, we observe a pattern:

[tex]\[ f(n+1) = -3 \times f(n) \][/tex]

Therefore, the recursive formula that can be used to generate the sequence is:
[tex]\[ f(n+1) = -3 f(n) \][/tex]

So the correct choice is:
[tex]\[ \boxed{f(n+1)=-3 f(n)} \][/tex]
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