What is the variance of the first 9 numbers of the Fibonacci sequence {0, 1, 1, 2, 3, 5, 8, 13, 21}?
Answers
Answer:
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Explanation:
A tiling with squares whose side lengths are successive Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 and 21.
In mathematics, the Fibonacci numbers, commonly denoted Fn, form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,[1]
{\displaystyle F_{0}=0,\quad F_{1}=1,}{\displaystyle F_{0}=0,\quad F_{1}=1,}
and
{\displaystyle F_{n}=F_{n-1}+F_{n-2},}{\displaystyle F_{n}=F_{n-1}+F_{n-2},}
for n > 1.
The beginning of the sequence is thus:
{\displaystyle 0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\;\ldots }{\displaystyle 0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\;\ldots }[2]
In some older books, the value {\displaystyle F_{0}=0}F_{0}=0 is omitted, so that the sequence starts with {\displaystyle F_{1}=F_{2}=1,}{\displaystyle F_{1}=F_{2}=1,} and the recurrence {\displaystyle F_{n}=F_{n-1}+F_{n-2}}{\displaystyle F_{n}=F_{n-1}+F_{n-2}} is valid for n > 2.[3][4]
The Fibonacci spiral: an approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling; (see preceding image)
Fibonacci numbers are strongly related to the golden ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases.
Fibonacci numbers are named after Italian mathematician Leonardo of Pisa, later known as Fibonacci. In his 1202 book Liber Abaci, Fibonacci introduced the sequence to Western European mathematics,[5] although the sequence had been described earlier in Indian mathematics,[6][7][8] as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths.
Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems.
Variance of the first 9 numbers of the sequence
Given:
Variance of the first 9 numbers of the sequence
To Find:
The variance of the first 9 numbers of the sequence zero, 1, 1, 2, 3, 5, 8, 13, 21
Solution:
A number ought to adjust this sequence of numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
This sequence will be generated by victimization the formula below:
Fibonacci Numbers Formula
F0 = 0, F1 = 1
and
Fn = Fn - 2 + Fn - 1
for n > 1.
List of the first nine fibonacci numbers
Fn or F(n)
Substitute the worth of n that is zero,
F0 = 0
Same here, substitute the worth of n = one
F1 = 1
F2 = 1
F3 = 2
F4 = F2 + F3 =3
Same here,
Substitute the worth of n,
We get,
F5 = F3 + F4 =5
F6 = F4 + F5 =8
F7 = F5 + F6 = thirteen
F8 = F6 + F7 = twenty one
Which is required variance of the sequence
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