Modified Fibonacci Sequence

Fibonacci Sequence

The genuine Fibonacci Sequence is defined by the linear recurrence equation:

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

Which goes like 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... In another words, the next number in the sequence is equal to the sum of its two predecessors.

The ratio of successive Fibonacci number tends rapidlly to a constant 1.61.

$\displaystyle\lim_{n\to\infty}\frac{F_{n}}{F_{n-1}}=\frac{1+\sqrt{5}}{2}\approx 1.61$

Extend for Mathematical Prove

It is known that

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

So that

$\frac{F_{n}}{F_{n-1}}=1+\frac{F_{n-2}}{F_{n-1}}$

Let

$r_{n}=\frac{F_{n}}{F_{n-1}}$

Hence

$r_{n}=1+\frac{1}{r_{n-1}}$

When n approaches to infinity

$\lim_{n\to\infty}\frac{F_{n}}{F_{n-1}}=\lim_{n\to\infty}r_{n}=L$

Then

$L=1+\frac{1}{L}$

Solve the equation

$L=\frac{1\pm\sqrt{5}}{2}$

Sence the ratio is positive

$L=\frac{1+\sqrt{5}}{2}$