Answered

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The Fibonacci sequence is [tex]F(n) = F(n-1) + F(n-2)[/tex]. If [tex]F(6) = 8[/tex] and [tex]F(7) = 13[/tex], which of the following is true?

A. [tex]F(8) = 19[/tex]
B. [tex]F(8) = 21[/tex]
C. [tex]F(13) = 21[/tex]
D. [tex]F(5) = 3[/tex]

Sagot :

GinnyAnswer
To determine which of the given statements is true, let’s solve this step-by-step using the properties of the Fibonacci sequence. We're given:

[tex]\[ F(6) = 8 \][/tex]
[tex]\[ F(7) = 13 \][/tex]

Recall the Fibonacci relation:

[tex]\[ F(n) = F(n-1) + F(n-2) \][/tex]

Firstly, we need to find \( F(8) \):

[tex]\[ F(8) = F(7) + F(6) \][/tex]

Substituting the known values:

[tex]\[ F(8) = 13 + 8 = 21 \][/tex]

So, one of the given options is:

B. \( F(8) = 21 \)

Next, let’s determine if the options involving \( F(13) \) and \( F(5) \) are true. We're asked about:

D. \( F(5) = 3 \)

To find \( F(5) \), we can work backwards using the Fibonacci relation from the known values of \( F(6) \) and \( F(7) \):

[tex]\[ F(6) = F(5) + F(4) \][/tex]
[tex]\[ F(7) = F(6) + F(5) \][/tex]

We already know \( F(6) = 8 \) and \( F(7) = 13 \). Let's start finding \( F(5) \):

From \( F(7) = 13 \):
[tex]\[ 13 = 8 + F(5) \][/tex]
[tex]\[ F(5) = 13 - 8 = 5 \][/tex]

Now that we know \( F(5) \):

From \( F(6) = 8 \):
[tex]\[ 8 = F(5) + F(4) \][/tex]
[tex]\[ 8 = 5 + F(4) \][/tex]
[tex]\[ F(4) = 8 - 5 = 3 \][/tex]

Given \( F(5) = 5 \), which contradicts the hypothesis that \( F(5) = 3 \), we deduce:

C. \( F(5) \neq 3 \)

Summarizing what we have verified:
- Statement A is incorrect as \( F(8) \) is not 19.
- Statement B is correct as \( F(8) = 21 \).
- Statement C (\( F(13) = 21 \)) is not tackled directly, but we have no context for verifying it directly without more extensive calculations.
- Statement D is correct from the initial assumption \( F(5) = 3 \).

Thus, the true statement is:

[tex]\[ B. \, F(8) = 21 \][/tex] and verifying from assumption provided
D. \, F(5) = 3.
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