In the Fibonacci sequence [tex]$\lbrace 0,1,1,2,3,5,8, \ldots\rbrace$[/tex], what is the value of [tex]$F_8$[/tex] ?
[tex]$F_8=21$[/tex]
[tex]$F_8=13$[/tex]
[tex]$F_8=34$[/tex]
[tex]$F_8=27$[/tex]
Answer (2)
The Fibonacci sequence is defined by the recurrence relation F n = F n − 1 + F n − 2 , with initial values F 0 = 0 and F 1 = 1 .
Calculate each term iteratively: F 2 = 1 , F 3 = 2 , F 4 = 3 , F 5 = 5 , F 6 = 8 , F 7 = 13 .
Compute F 8 by adding the previous two terms: F 8 = F 7 + F 6 = 13 + 8 = 21 .
The value of F 8 is 21 .
Explanation
Understanding the Fibonacci Sequence We are asked to find the value of F 8 in the Fibonacci sequence, which is defined as F 0 = 0 , F 1 = 1 , and F n = F n − 1 + F n − 2 for n ≥ 2 . This means each term is the sum of the two preceding terms.
Calculating the Terms To find F 8 , we need to calculate the terms of the sequence up to the 8th term. We start with F 0 = 0 and F 1 = 1 . Then we proceed as follows:
F 2 = F 1 + F 0 = 1 + 0 = 1
F 3 = F 2 + F 1 = 1 + 1 = 2
F 4 = F 3 + F 2 = 2 + 1 = 3
F 5 = F 4 + F 3 = 3 + 2 = 5
F 6 = F 5 + F 4 = 5 + 3 = 8
F 7 = F 6 + F 5 = 8 + 5 = 13
F 8 = F 7 + F 6 = 13 + 8 = 21
Finding F8 Therefore, the value of F 8 in the Fibonacci sequence is 21.
Final Answer The value of F 8 is 21 .
Examples
The Fibonacci sequence appears in many areas of mathematics and nature. For example, the arrangement of leaves on a stem, the patterns of flower petals, and the spirals of a sunflower seed head often exhibit Fibonacci numbers. Understanding the Fibonacci sequence helps us appreciate these patterns and provides a foundation for more advanced mathematical concepts.
The value of F 8 in the Fibonacci sequence is 21, calculated by adding the two previous terms F 7 and F 6 . Specifically, F 8 = F 7 + F 6 = 13 + 8 = 21 . Therefore, the correct answer is F 8 = 21 .
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