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FibonacciBlocks

Geometric representation of the Fibonacci numbers

The Fibonacci sequence is a recursive sequence, defined by

The sequence can then be written as

Properties[]

where is the golden ratio.

Proof[]

We are given this recurrence relation,

Which is subject to and . One may form an auxiliary equation in accordingly and solve for .

Through the use of the quadratic formula, one will obtain,

or equivalently

So we have,

where and are constants to be determined. Substituting the values we have

Solving for both variables, we obtain,

So, one has

as required.

Sum[]

For all integers ,

Proof[]

Proposition: given

as defined above,

Let ,

Therefore the proposition holds for . Assume that the proposition holds for . We may now make use of the inductive step. Let .

.

We know that

from the assumption that the proposition holds for .

So, we have,

Using the definition,

one obtains

which obeys the proposition

As the proposition holds for , and , the proposition holds for all natural numbers.

Binet's Formula[]

Binet's Formula is a theorem that allows one to determine  , where represents the Fibonacci Number.

The theorem is as follows:

Lucas sequence[]

The recursive effect from the Fibonacci sequence can also be applied with other starting numbers, like the Lucas numbers, which start with 2 and 1, instead of 0 and 1.

Trivia[]

  • Fibonacci numbers are claimed to be common in nature; for example, the shell of a nautilus being a Fibonacci spiral. However, this has been disputed with the spiral having a ratio measured between 1.24 to 1.43.[1]

References[]

  1. Peterson, Ivars (April 1, 2005). "Sea Shell Spirals". Science News. https://www.sciencenews.org/article/sea-shell-spirals. 

Terms[]

Input Fibonacci Lucas
0 0 2
1 1 1
2 1 3
3 2 4
4 3 7
5 5 11
6 8 18
7 13 29
8 21 47
9 34 76
10 55 123
11 89 199
12 144 322
13 233 521
14 377 843
15 610 1364
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