Question

Show that $r\, = \,\dfrac{{1 + \sqrt 5 }}{2}$for the Fibonacci sequence?

Answer 1

Hint:This value of $r\, = \,\dfrac{{1 + \sqrt 5 }}{2}$is called the golden ratio or golden number. We have many applications of golden ratio in nature. Sequence of number 0,1,1,2,3,5,8,13,… is called Fibonacci Sequence and the number of this series is Fibonacci Number. To get this value of $r$ we will use methods of root of a quadratic equation and definition of Fibonacci sequence.

Complete step by step solution:
Lets us see, how do we get$r\, = \,\dfrac{{1 + \sqrt 5 }}{2}$ from the
Fibonacci sequence?

Before proving this we will write the recurrence relation of the Fibonacci sequence and then convert it to a quadratic equation. Solve the root of the quadratic equation using the quadratic formula to get$r\, = \,\dfrac{{1 + \sqrt 5 }}{2}$.

Definition: Fibonacci series is defined recursively by this recurrence relation.
$
{F_0} = 0 \\

{F_1} = 1 \\

{F_n} = {F_{n - 1}} + {F_{n - 2}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\forall \,n = 2) \\
$

If we suppose that the general term of Fibonacci sequence is${F_n} = a{r^n}$. So our recurrence relation of Fibonacci sequence will become
$a{r^n}\, = \,a{r^{n - 1}} + a{r^{n - 2}}\,\,\,\,\,\,\,\,\,\,\,\,(\forall \,n = 2)$

Taking out $a{r^{n - 2}}$from both side of the above equation we get
${r^2}\, = \,r\, + \,1$

${r^2} - r - 1\, = \,0$

Now, applying the quadratic formula we find the root of the quadratic equation${r^2} - r - 1\, = \,0$. We get roots as follows:

$
a = 1 \\

b = - 1 \\

c = - 1 \\
$

$
r = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} \\

r = \dfrac{{1 \pm \sqrt {{{( - 1)}^2} - 4 \cdot 1 \cdot ( - 1)} }}{{2 \cdot 1}} \\

r = \dfrac{{1 \pm \sqrt {1 + 4} }}{2} \\

r = \dfrac{{1 \pm \sqrt 5 }}{2} \\
$

Hence we get the root of quadratic equation ${r^2} - r - 1\, = \,0$which we got from the recurrence relation of the Fibonacci sequence is $r = \dfrac{{1 + \sqrt 5 }}{2}$ and$r = \dfrac{{1 - \sqrt 5 }}{2}$.

The number $r = \dfrac{{1 + \sqrt 5 }}{2}$ is called the golden ratio or golden number.

Note: General term of Fibonacci sequence is of the form ${F_n}\, = \,A\left( {\dfrac{{1 + \sqrt 5 }}{2}} \right)\, + B\left( {\dfrac{{1 - \sqrt 5 }}{2}} \right)$ and we get the value of constant A and B from initial condition of recurrence of the Fibonacci series which is${F_0} = 0\,\,\,\,and\,\,\,\,{F_1} = 1$. We can also find the roots of quadratic also using the completing square and factor method. Similarly, we can also find the general term of Lucas series.