FANDOM


FibonacciBlocks

Geometric representation of the Fibonacci numbers

The Fibonacci sequence is a recursive sequence, defined as

a_0=0, a_1=1
\{a_i\}_{k=0}^{\infty} = a_{i-1} + a_{i-2} = 0, 1, 1, 2, 3, 5, 8, 13, 21,  ...

Properties

\lim_{n \to \infty} \frac{a_{n + 1}}{a_n} = \phi

\phi is equal to the golden ratio.

a_n = \frac{\phi^n - (1-\phi)^n}{\sqrt 5 } = \frac{(1+\sqrt 5)^n - (1-\sqrt 5)^n}{ 2^n \sqrt 5 }

Proof

We are given this recurrence relation,

F_{n} = F_{n-1}+F_{n-2}

Which is subject to F_{1} = 1 and F_{2} = 1. One may form an auxiliary equation in \lambda accordingly and solve for \lambda.

\lambda^2 - \lambda - 1 = 0

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

\lambda = \frac{1+\sqrt{5}}{2} or \lambda = \frac{1-\sqrt{5}}{2}

Or otherwise,

\lambda = \phi or \lambda = 1 - \phi

So we have,

F_{n} = C_1(\phi)^n + C_2(1-\phi)^n

Where C_1 and C_2 are constants to be determined. Substituting the values we have

1 = C_1\phi + C_2(1-\phi)
1 = C_1\phi^2 + C_2(1-\phi)^2

Solving for both variables, we obtain,

C_1 = \frac{1}{\sqrt{5}}
C_2 = -\frac{1}{\sqrt{5}}

So, one has

F_n = \frac{1}{\sqrt{5}} \left(\phi^n - [1-\phi]^n\right)

As required \blacksquare

Sum

For all integers n greater than or equal to 1,

\sum_{t=1}^{n} F_{t} = F_{n+2} - 1

Proof

Proposition: given F_{n} as defined above,

\sum_{t=1}^{n} F_{t} = F_{n+2} - 1 \forall n \in \mathbb{N}

Let n = 1,

\sum_{t=1}^{n} F_{t} = F_{1} = 1 = F_{3} - 1

Therefore the proposition holds for n=1. Assume that the proposition holds for n=\lambda. We may now make use of the inductive step. Let n=\lambda + 1.

\sum_{t=1}^{\lambda+1} F_{t} = \sum_{t=1}^{\lambda} F_{t} + F_{\lambda + 1}.

We know that \sum_{t=1}^{\lambda} F_{t} = F_{\lambda + 2} - 1, from the assumption that the proposition holds for n = \lambda.

So, we have,

\sum_{t=1}^{\lambda+1} F_{t} = F_{\lambda + 2} + F_{\lambda + 1} - 1

Using the definition,

F_{n} = F_{n-1} + F_{n-2}

One obtains

\sum_{t=1}^{\lambda+1} F_{t} = F_{\lambda + 3} - 1

Which obeys the proposition

\sum_{t=1}^{\lambda} F_{t} = F_{\lambda + 2} - 1.

As the proposition holds for n=1, n = \lambda and n = S(\lambda), the proposition holds for all natural numbers. \blacksquare

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. 

Ad blocker interference detected!


Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.