$$
\boxed {A_{n+2}=A_{n+1}+A_n } ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.1)
$$
That is, each term is equal to the sum of the two previous terms. This sequence requires two seed values, $A_0$ and $A_1$. The simple case of $A_0=0$ and $A_1=1$ may be considered as an example. This gives the sequence
$$
\boxed{0,1,1,2,3,5,8,13,21,34,55,89,144, \ldots} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.2)
$$
This sequence can be
extended indefinitely by applying the recursion relation. It may also be
extended to negative values of the index, $n$, by applying a recursion relation
based on Eq. (2.1) to the values given in Eq. $(2.2)$ yielding a sequence which
extends indefinitely in both directions;
$$
\boxed{\ldots 34,-21,13,-8,5,-3,2,-1,1,0,1,1,2,3,5,8, \ldots} .~~~~~~~~~~~~~~~~~~~~~~~~(2.3)
$$
In this particular case the values of the terms with negative indices are numerically the same as the corresponding terms with positive indices but they alternate in sign. This is an interesting property of this particular additive sequence which will be discussed further in Chapter 5 , although it is not a property of additive sequences in general.
Another simple numerical sequence, referred to as the geometric sequence, is generated by the recursion relation
$$
\boxed{A_{n+1}=\alpha A_n \text {. }}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (2.4)
$$
That is, each term is the previous term multiplied by some constant factor. This sequence may be generated on the basis of one seed value and the value of the constant factor. A simple example uses $A_0=1$ and $\alpha=2$. This gives the familiar sequence of powers of 2 ;
$$
\boxed{1,2,4,8,16,32,64,128,256,512, \ldots .} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.5)
$$
Again it is straightforward to extend this sequence to negative values of the indices;
$$
\text { ... } \frac{1}{32}, \frac{1}{16}, \frac{1}{8}, \frac{1}{4}, \frac{1}{2}, 1,2,4, \ldots . ~~~~~(2.6)
$$
A comparison of Eqs. (2.3) and (2.6) would seem to illustrate the fundamental differences between additive and geometric sequences. However, these differences are the result of the particular choice of the multiplicative constant in Eq. $(2,4)$. Different choices for this quantity can yield very different results. Consider, for example, the possibility that a sequence could be both additive and geometric; that is, the terms would satisfy both Eq. (2.1) and Eq. (2.4). These two equations can be combined to give the constraining relations for $\alpha$. From Eq. (2.4) we can write
$$
A_{n+2}=\alpha A_{n+1}=\alpha^2 A_n .~~~~~~~~~~~~~~~~~~~~~~~~~(2.7)
$$
Eqs. (2.1) and $(2,7)$ yield the relation
$$
\alpha^2 A_n=\alpha A_n+A_n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.8)
$$
or simply
$$
\alpha^2-\alpha-1=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.9)
$$
This equation is known as the Fibonacci quadratic equation and is easily solved to yield the two roots
$$
\alpha_1=\frac{1+\sqrt{5}}{2}=r ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.10)
$$
and
$$
\alpha_2=\frac{1-\sqrt{5}}{2}=-\frac{1}{\tau}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.11)
$$
It is straightforward to construct a geometric sequence using the value of $\alpha_1$ as the constant factor and a seed value of (say) $A_0=1$. This gives
$$
1, \tau, \tau^2, \tau^3, r^4, \tau^5, \ldots~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.12)
$$
Extending this to negative indices yields
$$
\ldots \tau^{-3}, \tau^{-2}, \tau^{-1}, 1, \tau, \tau^2, \tau^3, \ldots ~~~~~~~~~~~~~~~~~~~(2.13)
$$
$$
\boxed{\ldots 34,-21,13,-8,5,-3,2,-1,1,0,1,1,2,3,5,8, \ldots} .~~~~~~~~~~~~~~~~~~~~~~~~(2.3)
$$
In this particular case the values of the terms with negative indices are numerically the same as the corresponding terms with positive indices but they alternate in sign. This is an interesting property of this particular additive sequence which will be discussed further in Chapter 5 , although it is not a property of additive sequences in general.
Another simple numerical sequence, referred to as the geometric sequence, is generated by the recursion relation
$$
\boxed{A_{n+1}=\alpha A_n \text {. }}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (2.4)
$$
That is, each term is the previous term multiplied by some constant factor. This sequence may be generated on the basis of one seed value and the value of the constant factor. A simple example uses $A_0=1$ and $\alpha=2$. This gives the familiar sequence of powers of 2 ;
$$
\boxed{1,2,4,8,16,32,64,128,256,512, \ldots .} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.5)
$$
Again it is straightforward to extend this sequence to negative values of the indices;
$$
\text { ... } \frac{1}{32}, \frac{1}{16}, \frac{1}{8}, \frac{1}{4}, \frac{1}{2}, 1,2,4, \ldots . ~~~~~(2.6)
$$
A comparison of Eqs. (2.3) and (2.6) would seem to illustrate the fundamental differences between additive and geometric sequences. However, these differences are the result of the particular choice of the multiplicative constant in Eq. $(2,4)$. Different choices for this quantity can yield very different results. Consider, for example, the possibility that a sequence could be both additive and geometric; that is, the terms would satisfy both Eq. (2.1) and Eq. (2.4). These two equations can be combined to give the constraining relations for $\alpha$. From Eq. (2.4) we can write
$$
A_{n+2}=\alpha A_{n+1}=\alpha^2 A_n .~~~~~~~~~~~~~~~~~~~~~~~~~(2.7)
$$
Eqs. (2.1) and $(2,7)$ yield the relation
$$
\alpha^2 A_n=\alpha A_n+A_n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.8)
$$
or simply
$$
\alpha^2-\alpha-1=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.9)
$$
This equation is known as the Fibonacci quadratic equation and is easily solved to yield the two roots
$$
\alpha_1=\frac{1+\sqrt{5}}{2}=r ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.10)
$$
and
$$
\alpha_2=\frac{1-\sqrt{5}}{2}=-\frac{1}{\tau}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.11)
$$
It is straightforward to construct a geometric sequence using the value of $\alpha_1$ as the constant factor and a seed value of (say) $A_0=1$. This gives
$$
1, \tau, \tau^2, \tau^3, r^4, \tau^5, \ldots~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.12)
$$
Extending this to negative indices yields
$$
\ldots \tau^{-3}, \tau^{-2}, \tau^{-1}, 1, \tau, \tau^2, \tau^3, \ldots ~~~~~~~~~~~~~~~~~~~(2.13)
$$
Using the seed values of $A_0=1$ and $A_1=\tau$ from Eq. (2.12) a corresponding additive sequence may be constructed using the recursion relation of Eq. (2.1). For negative and positive indices this sequence is
$$
-3 r+5,2 \tau-3,-\tau+2, \tau-1,1, \tau, \tau+1,2 \tau+1,3 r+2, \ldots . ~~~~~~~~~~~~~~~~(2.14)
$$
Numerically the terms in this sequence are the same as those in the geometric sequence in Eq. $(2,13)$. These terms may be equated to yield some interesting relationships between powers of $\tau$ and linear expressions in $\tau$. Some of these are
$$\boxed{
\begin{gathered}
2 \tau-3&=\tau^{-3} \\
-\tau+2&=\tau^{-2} \\
\tau-1&=\tau^{-1} \\
1&=1 \\
\tau&=\tau \\
\tau+1&=\tau^2 \\
2 \tau+1&=\tau^3 .
\end{gathered}} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.15)
$$
In general, powers of the golden ratio may be expressed as
Table 2.1. Some coefficients and exponents in the relationship given by Eq. $(2,16)$.
\begin{array}{|c|c|c|}
\hline n & a_n & a_{n-1} \\
\hline-8 & -21 & 34 \\
-7 & 13 & -21 \\
-6 & -8 & 13 \\
-5 & 5 & -8 \\
-4 & -3 & 5 \\
-3 & 2 & -3 \\
-2 & -1 & 2 \\
-1 & 1 & -1 \\
0 & 0 & 1 \\
1 & 1 & 0 \\
2 & 1 & 1 \\
3 & 2 & 1 \\
4 & 3 & 2 \\
5 & 5 & 3 \\
6 & 8 & 5 \\
7 & 13 & 8 \\
8 & 21 & 13 \\
\hline
\end{array}
$$
\boxed{a_n \tau+a_{n-1}=\tau^n } ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.16)
$$
where the coefficients, $a_n$, as given in Table 2.1 are the $A_n$ of the additive sequence in Eq. (2.3). This relationship is discussed further in Chapter 5.
Another sequence which is both additive and geometric can be derived using the other root of the quadratic equation as given by Eqs. (2.11) and (2.15), $\alpha_2=-\tau^{* 1}=1-\tau$. This gives the sequence
$$
\cdots-\tau^3, \tau^2,-\tau, 1,-\tau^{-1}, \tau^{-2},-\tau^{-3} \ldots~~~~~~~~~~~~~(2.17)
$$
and the corresponding sequence based on the additive recursion relation is found to be
$$
\ldots-3-\frac{2}{\tau}, 2+\frac{1}{\tau},-1-\frac{1}{\tau}, 1,-\frac{1}{\tau}, 1-\frac{1}{\tau}, 1-\frac{2}{\tau}, 2-\frac{3}{\tau}, \ldots .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.18)
$$
Equating terms from Eqs. $(2,17)$ and $(2,18)$ allows for the derivation of relations of the form
$$
a_{n+1}+\frac{a_n}{\tau}=\tau^n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.19)
$$
where the coefficients are again the terms in the additive sequence of Eq. (2.3). It can be shown that these expressions are algebraically equivalent to those of Eq. (2.16) by multiplying both sides of Eq. $(2.19)$ by $\tau$.
The above discussion concerning numerical sequences illustrates the relationship of the golden ratio to some fundamental properties of numbers. Additional insight into the properties of the golden ratio may be gained by taking a somewhat more geometric approach. In fact, it is this occurrence of the golden ratio which is responsible for its appeal to the ancient philosophers and for the derivation of its name; the golden ratio. Consider a line $A C$ which is divided by a point $B$ as illustrated in Fig. 2.1 in such a way that the ratio of the lengths of the two segments is the same as the ratio of the length of the longer segment to the entire line. If the length $A B$ is arbitrarily set equal to 1 and the length of the total line is called $x$ then the segment $B C=x-1$ and the ratios of lengths may be expressed as
$$
\frac{x}{1}=\frac{1}{x-1}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.20)
$$
x^2-x-1=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.21)
$$
This is the Fibonacci equation which has the roots given in terms of the golden ratio by Eqs. (2.10) and $(2.11) ; \tau$ and $-1 / \tau$. Obviously it is the positive root which has some physical significance in the context of this problem. Alternately the total length of the line may be set to 1 and segment $A B$ may be arbitrarily called $x$. The ratios are then
$$
\frac{1}{x}=\frac{x}{1-x}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.22)
$$
$$
-3 r+5,2 \tau-3,-\tau+2, \tau-1,1, \tau, \tau+1,2 \tau+1,3 r+2, \ldots . ~~~~~~~~~~~~~~~~(2.14)
$$
Numerically the terms in this sequence are the same as those in the geometric sequence in Eq. $(2,13)$. These terms may be equated to yield some interesting relationships between powers of $\tau$ and linear expressions in $\tau$. Some of these are
$$\boxed{
\begin{gathered}
2 \tau-3&=\tau^{-3} \\
-\tau+2&=\tau^{-2} \\
\tau-1&=\tau^{-1} \\
1&=1 \\
\tau&=\tau \\
\tau+1&=\tau^2 \\
2 \tau+1&=\tau^3 .
\end{gathered}} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.15)
$$
In general, powers of the golden ratio may be expressed as
Table 2.1. Some coefficients and exponents in the relationship given by Eq. $(2,16)$.
\begin{array}{|c|c|c|}
\hline n & a_n & a_{n-1} \\
\hline-8 & -21 & 34 \\
-7 & 13 & -21 \\
-6 & -8 & 13 \\
-5 & 5 & -8 \\
-4 & -3 & 5 \\
-3 & 2 & -3 \\
-2 & -1 & 2 \\
-1 & 1 & -1 \\
0 & 0 & 1 \\
1 & 1 & 0 \\
2 & 1 & 1 \\
3 & 2 & 1 \\
4 & 3 & 2 \\
5 & 5 & 3 \\
6 & 8 & 5 \\
7 & 13 & 8 \\
8 & 21 & 13 \\
\hline
\end{array}
$$
\boxed{a_n \tau+a_{n-1}=\tau^n } ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.16)
$$
where the coefficients, $a_n$, as given in Table 2.1 are the $A_n$ of the additive sequence in Eq. (2.3). This relationship is discussed further in Chapter 5.
Another sequence which is both additive and geometric can be derived using the other root of the quadratic equation as given by Eqs. (2.11) and (2.15), $\alpha_2=-\tau^{* 1}=1-\tau$. This gives the sequence
$$
\cdots-\tau^3, \tau^2,-\tau, 1,-\tau^{-1}, \tau^{-2},-\tau^{-3} \ldots~~~~~~~~~~~~~(2.17)
$$
and the corresponding sequence based on the additive recursion relation is found to be
$$
\ldots-3-\frac{2}{\tau}, 2+\frac{1}{\tau},-1-\frac{1}{\tau}, 1,-\frac{1}{\tau}, 1-\frac{1}{\tau}, 1-\frac{2}{\tau}, 2-\frac{3}{\tau}, \ldots .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.18)
$$
Equating terms from Eqs. $(2,17)$ and $(2,18)$ allows for the derivation of relations of the form
$$
a_{n+1}+\frac{a_n}{\tau}=\tau^n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.19)
$$
where the coefficients are again the terms in the additive sequence of Eq. (2.3). It can be shown that these expressions are algebraically equivalent to those of Eq. (2.16) by multiplying both sides of Eq. $(2.19)$ by $\tau$.
The above discussion concerning numerical sequences illustrates the relationship of the golden ratio to some fundamental properties of numbers. Additional insight into the properties of the golden ratio may be gained by taking a somewhat more geometric approach. In fact, it is this occurrence of the golden ratio which is responsible for its appeal to the ancient philosophers and for the derivation of its name; the golden ratio. Consider a line $A C$ which is divided by a point $B$ as illustrated in Fig. 2.1 in such a way that the ratio of the lengths of the two segments is the same as the ratio of the length of the longer segment to the entire line. If the length $A B$ is arbitrarily set equal to 1 and the length of the total line is called $x$ then the segment $B C=x-1$ and the ratios of lengths may be expressed as
$$
\frac{x}{1}=\frac{1}{x-1}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.20)
$$
or
$$x^2-x-1=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.21)
$$
This is the Fibonacci equation which has the roots given in terms of the golden ratio by Eqs. (2.10) and $(2.11) ; \tau$ and $-1 / \tau$. Obviously it is the positive root which has some physical significance in the context of this problem. Alternately the total length of the line may be set to 1 and segment $A B$ may be arbitrarily called $x$. The ratios are then
$$
\frac{1}{x}=\frac{x}{1-x}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.22)
$$
or
$$
x^2+x-1=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.23)
$$
This quadratic equation has roots which may be expressed in terms of the golden ratio as
$$
x_1=\frac{\sqrt{5}-1}{2}=\frac{1}{\tau}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.24)
$$
x^2+x-1=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.23)
$$
This quadratic equation has roots which may be expressed in terms of the golden ratio as
$$
x_1=\frac{\sqrt{5}-1}{2}=\frac{1}{\tau}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.24)
$$
and
$$
x_2=-\frac{\sqrt{5}+1}{2}=-t~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.25)
$$
Again only the positive root has physical significance and shows the ratio of lengths to be related to the golden ratio.
Some interesting mathematical relationships involving the golden ratio can be derived by combining powers of $\tau$. For example, a simple inspection of relationships such as those shown in Eq. (2.15) and Table 2.1, will allow for the derivation of expressions involving both positive and negative powers of the golden ratio. The simplest of these is
$$
r^n+(-1)^n r^{-n}=L_n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.26)
$$
where $L_n$ is an integer that takes on values $L_n=1,3,4,7,11,18, \ldots$ for $n=1,2,3,4,5$, $6, \ldots$. These are the so-called Lucas numbers and are disussed further in Chapter 6 . This expression is somewhat remarkable as it shows that the sum of two irrational numbers can be equal to a rational number.
Another interesting relationship involving the golden ratio may be obtained directly from the Fibonacci quadratic equation, Eq. (2.9). This may be written for $r$ as
$$
\tau=\sqrt{1+\tau}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.27)
$$
Substituting the left hand side for $\tau$ in the square root on the right hand side gives
$$
t=\sqrt{1+\sqrt{1+\tau}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.28)
$$
This procedure may be continued indefinitely to yield
$$
\tau=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\ldots}}}}}~~~~~~~~~~~~~~~~~~~~~~~(2.29)
$$
Along similar lines it is known that the positive root of Eq. $(2.23)$ is $1 / \tau$. This expression may be rearranged and the substitution for the term in the square root performed indefinitely to give
$$
\frac{1}{\tau}=\sqrt{1-\sqrt{1-\sqrt{1-\sqrt{1-\sqrt{1-\ldots}}}}} ~~~~~~~~~~~~~~~~~~(2.30)
$$
The expression in Eq. (2.30) provides one means of calculating the golden ratio to a high degree of accuracy using a computer. It is, however, less time consuming to calculate $\tau$ directly on the basis of Eq. $(2.10)$ by first calculating the square root of 5. An irrational square root can be c alculated to a n a rbitrary a ccuracy u sing a simple iterative technique. To calculate a square root to an accuracy of $N$ digits requires a number of basic arithmetic operations which is proportional to $N^2$. An early report of the use of a computer to calculate the golden ratio to high accuracy provided $\tau$ to 4599 decimal places; see Berg (1966). This required about 20 minutes on an IBM 1401 main frame computer. Today this calculation can be done on an IBM Pentium personal computer in about 2 seconds. It is straightforward to determine the validity of the calculated values. One method is to substitute the calculated value of $\tau$ into the Fibonacci equation (Eq. (2.9)) and perform the operations to the required number of decimal places and show that the identity holds. An equivalent method is to calculate the reciprocal of $\tau$ and show that $1 / \tau=\tau-1$ holds to the required accuracy.
Source :
The Golden Ratio and Fibonacci Number by. Richard A.Dunlap, page 7-13
x_2=-\frac{\sqrt{5}+1}{2}=-t~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.25)
$$
Again only the positive root has physical significance and shows the ratio of lengths to be related to the golden ratio.
Some interesting mathematical relationships involving the golden ratio can be derived by combining powers of $\tau$. For example, a simple inspection of relationships such as those shown in Eq. (2.15) and Table 2.1, will allow for the derivation of expressions involving both positive and negative powers of the golden ratio. The simplest of these is
$$
r^n+(-1)^n r^{-n}=L_n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.26)
$$
where $L_n$ is an integer that takes on values $L_n=1,3,4,7,11,18, \ldots$ for $n=1,2,3,4,5$, $6, \ldots$. These are the so-called Lucas numbers and are disussed further in Chapter 6 . This expression is somewhat remarkable as it shows that the sum of two irrational numbers can be equal to a rational number.
Another interesting relationship involving the golden ratio may be obtained directly from the Fibonacci quadratic equation, Eq. (2.9). This may be written for $r$ as
$$
\tau=\sqrt{1+\tau}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.27)
$$
Substituting the left hand side for $\tau$ in the square root on the right hand side gives
$$
t=\sqrt{1+\sqrt{1+\tau}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.28)
$$
This procedure may be continued indefinitely to yield
$$
\tau=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\ldots}}}}}~~~~~~~~~~~~~~~~~~~~~~~(2.29)
$$
Along similar lines it is known that the positive root of Eq. $(2.23)$ is $1 / \tau$. This expression may be rearranged and the substitution for the term in the square root performed indefinitely to give
$$
\frac{1}{\tau}=\sqrt{1-\sqrt{1-\sqrt{1-\sqrt{1-\sqrt{1-\ldots}}}}} ~~~~~~~~~~~~~~~~~~(2.30)
$$
The expression in Eq. (2.30) provides one means of calculating the golden ratio to a high degree of accuracy using a computer. It is, however, less time consuming to calculate $\tau$ directly on the basis of Eq. $(2.10)$ by first calculating the square root of 5. An irrational square root can be c alculated to a n a rbitrary a ccuracy u sing a simple iterative technique. To calculate a square root to an accuracy of $N$ digits requires a number of basic arithmetic operations which is proportional to $N^2$. An early report of the use of a computer to calculate the golden ratio to high accuracy provided $\tau$ to 4599 decimal places; see Berg (1966). This required about 20 minutes on an IBM 1401 main frame computer. Today this calculation can be done on an IBM Pentium personal computer in about 2 seconds. It is straightforward to determine the validity of the calculated values. One method is to substitute the calculated value of $\tau$ into the Fibonacci equation (Eq. (2.9)) and perform the operations to the required number of decimal places and show that the identity holds. An equivalent method is to calculate the reciprocal of $\tau$ and show that $1 / \tau=\tau-1$ holds to the required accuracy.
Source :
The Golden Ratio and Fibonacci Number by. Richard A.Dunlap, page 7-13
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