My father was previously a mathematics professor and my brother is involved in computer engineering. Needless to say, I was brought up in an environment filled with mathematics and computers. I too found great interest in these two subjects which encouraged me to explore their depths. Consequently, I had more knowledge than the other students of my class regarding these two subjects.
I was in the 6th grade when my brother encouraged me to write a computer program on the Fibonacci series. I had no idea about how to proceed. Later in the evening, I asked my brother to help. With his assistance, I managed to run the program successfully.
I rushed to my father to tell him about it. He had the habit of relating almost everything with mathematics. He at once asked me what I knew about the Fibonacci series with respect to mathematics.
I then realized that I have seen Arithmetic series, Geometric series and Harmonic series, but never before a Fibonacci series. I had absolutely no understanding about a Fibonacci series equation. Thus, I decided to try and find out myself.
In the process of my study, I came across the term, “golden ratio.” It surprised me to note that the Great Pyramid, hurricanes, cosmos, anatomies, and flower petals, all, maintain the golden ratio. Even the length and width of each DNA turn and Beethoven’s 5th symphony fit the golden ratio; if a musical note maintains the ratio, it becomes soothing to the ear.
This field proved to be more interesting than I ever expected and therefore it urged me to fortify my understanding of it. In this IA, I will derive the equation of nth term of a Fibonacci Sequence and represent it graphically for a better, clearer understanding.
The objective of this exploration is to derive a relation between the terms of a Fibonacci Sequence which facilitates the non-sequential calculation of a Fibonacci Sequence term from an antecedent term or a subsequent term.
What is the generalized scaling factor which should be multiplied with any term of the Fibonacci Sequence to get the next term?
In the Fibonacci Sequence, each term is the sum of previous two terms. Each term in this sequence can be expressed as:
T_{n} = T_{n - 1} + T_{n - 2} ..............(equation - 1)
Example of a Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 …
Let us plot a Fibonacci Sequence where the first two terms are taken as 0 and 1. Now, the graph is plotted taking the abscissa of each point as the present term (T_{n}) and ordinate of each point as the previous term (T_{n - 1}). The graphs have been plotted using online tool Desmos.
In Fig. 1, we have observed that the points of the sequence lie close to the straight line. However, in Fig. 2, it has been observed that as the value of n increases, the points of the sequence lie exactly (with negligible percentage of error) on the straight line. The equation of the straight line for the above-mentioned Fibonacci Series is given by:
y = 0.6181x - 0.0330268 ……………(equation - 2)
Since all the values of x and y are not the terms of the Fibonacci Sequence, this equation cannot be taken as the equation of Fibonacci Sequence.
If the Fibonacci Sequence is developed representing in a matrix form with the value of present term at position ROW 1 and COLUMN 1, and the value of the previous term at ROW 2 and COLUMN 1 in a [2 × 1] matrix, from the properties of Eigen Vectors, the above straight line (2) can be stated as the Eigen Vector of the Matrix. Using the Eigen Value, we will be able to find the next values of the sequence.
Then the matrix will be:
\(\biggl({T_n\over T_{n-1}}\biggl)\ =(1\ 1\ 1\ 0)\biggl({T_{n-1}\over T_{n-2}}\biggl)\)……………(equation - 3)
\(=> \biggl({x\over y}\biggl)=(1\ 1\ 1\ 0)\biggl({T_{n-1}\over T_{n-2}}\biggl) \).……………(equation - 4)
Here, the value of present term is denoted as x, and the value of the previous term is denoted as y.
A sequence is defined by:
u_{1} = 1,u_{n} = \({1\over 1+u_{n-1}}\){for n > 1}
The first five terms of this sequence are given as follows:
u_{1 }= 1
u_{2} = \( {1\over 1+u_1} \ ={1\over 1+1}\) = 0.5
u_{3} \({1\over 1+u_2} \ ={1\over 1+0.5}\) = 0.6667
u_{4} =\({1\over 1+u_3} \ ={1\over 1+0.6667}\) = 0.6000
u_{5} = \({1\over 1+u_4} \ ={1\over 1+0.6000}\) = 0.625
Value of all the subsequent terms are solved using a python code which is shown. The python code was simulated in PyCharm IDE.
From the above result, we can see that, 11^{th} term (u_{11}) onwards, the value of every term is approximately equal to 0.618.
From the graph and the values of each term as shown in Figure 3 and Figure 4, it can be noted that with increase in the number of terms n, the value of the term tends to become constant.
Eigen Vector of any matrix may be defined as a non-zero vector of a straight line which only gets scaled without any rotation when multiplied by the matrix.
The Eigen Vector \(\vec{ v}\) of any Matrix A may be represented as the equation stated below, where \(\lambda\) is the Eigen Value of the Eigen Vector.
\(\)
\(A \vec{ v}\ =\ \lambda \vec{ v}\)
\(=> A \vec{ v}\ =\ \lambda \vec{ v}\)……………………(equation - 6)
Here I is an identity matrix with the order of Matrix A.
Eigen Value of an Eigen Vector may be defined as the factor by which it is scaled.
From Equation (6), let us derive the formula of Eigen Value:
\(A\vec v \ = \lambda I \ \vec v\)
\( => A\vec v \ = \lambda I \ \vec v = 0\)
\( => [A \ - \lambda I] \ \vec v = 0\) ………………………(equation - 7)
From the definition of Eigen Vector, \(\vec v\) is a non-zero vector. So, we can write:
det det ( A - λI ) = 0 …………………………(equation - 8)
This is defined as the Characteristic Polynomial of a matrix.
In this section of the exploration, I will try to derive the equation of Fibonacci Sequence using the above equations (4), (7), and (8) which we have derived from the information discussed above.
\(\biggl({T_n\over T_{n-1}}\biggl) = (1\ 1\ 1\ 0)\biggl({T_{n-1}\over T_{n-2}}\biggl)\)
For simplicity, let us consider: (1 1 1 0 ) = X
\(∴\biggl({T_n\over T_{n-1}}\biggl)\ = X \biggl({T_{n-1}\over T_{n-2}}\biggl)\)
\(=>\bigg(\frac{T_n}{T_{n-1}}\bigg)=X\bigg\{X\bigg(\frac{T_{n-2}}{T_{n-3}}\bigg)\bigg\}=X^2\bigg(\frac{T_{n-2}}{T_{n-3}}\bigg)\)
\(=> \biggl({T_n\over T_{n-1}}\biggl)\ = X^{n-1} \bigg({T_{1}\over T_{0}}\bigg) = X^{n-1}\biggl({1\over 0 }\biggl) \)
So, we can see that by substituting the values of T for different values of n with the principal equation (4), we can derive a relationship of nth term with the first two terms.
In order to diagonalize the value of X rather than X ^{n-1}, let us consider two non-zero matrices M and K, such that:
X.M = M.K
=> X = M.K.M ^{-1}
=> K = M ^{- 1}.X.M ……………………(equation - 9)
Now, we will find the value of K using the characteristic equation of Eigen Vectors and Eigen Values.
From equation (8), we can write:
det det ( A - λI ) = 0
Here, A = X = (1 1 1 0 ) and I = (1 0 0 1)
∴det det {(1 1 1 0 ) - λ (1 0 0 1)} = 0
=> det det (1 - λ 1 1 - λ) = 0
=> (1 - λ)( - λ) - (1) (1) = 0
=> λ^{2} - λ - 1 = 0………………………(equation - 10)
Using Sreedhar Acharya’s Formula
\(\lambda_1 = {1+\sqrt{5}\over 2}\)
\(\lambda_2 = {1-\sqrt{5}\over 2}\)
From equation (7), we can write:
Case 1: Let, λ = λ_{1}
_{\(\big[X - \lambda_1I\big]\vec v_1\ = 0\)}
\(=> (1 - \lambda _1 11- \lambda_1)\biggl({x\over y}\biggl) = 0\)
\(=>\biggl({x -x\lambda_1 + y\over x-y\lambda_1}\biggl) = 0\)
From the above equation, we can write,
x - yλ_{1 }= 0
=> x = yλ_{1}
Let y = 1, then the value of \(x = \lambda_1\)
\(\vec v_1= \biggl({\lambda_1\over 1}\biggl)\)
Case 2: Let, λ = λ_{2}
_{\(\big[X - \lambda_2I\big]\vec v_2\ = 0\)}
\(=> (1 - \lambda _2 \ 1\ 1 - \lambda _2)({x\over y}) = 0\)
\(=> \biggl({x-x\lambda _2 + y \over x-y\lambda_2}\biggl)\ = 0\)
From the above equation, we can write,
x - yλ_{2 }= 0
=> x = yλ_{2}
Let y =1 then the value of \(x = \lambda_2.\)
\(\vec v_2 = \bigg({\lambda _2\over 1}\bigg)\)
Therefore, we can conclude that:
M = (v_{1},v_{2}) = (λ_{1} λ_{2} 1 1)
\(M^{-1}\ = {1\over detdet (\lambda_1 \lambda _2\ 1\ 1)}adj(\lambda_1 \lambda _2\ 1\ 1)\)
\(=> M^{-1}\ ={1\over \lambda _1 - \lambda_2}(1 -\lambda_2 -1 \lambda_1)\)
\(=> M^{-1}\ = {1\over {1+\sqrt{5}\over 2}{1-\sqrt{5}\over 2}} (1 - \lambda _2 - 1\lambda_1)\)
\(=> M^{-1}\ = {1\over \sqrt{5}} (1 - \lambda _2 - 1\lambda_1)\)
From equation (9), we can write:
K = M^{-1}.X.M
\(=> K ={1\over \sqrt{5}} (1-\lambda_2 - 1\lambda_1) . (1\ 1\ 1\ 0). (\lambda_1\lambda_2\ 1\ 1)\)
\(=> K ={1\over \sqrt{5}} (1-\lambda_2 - 1\lambda_1) . (\lambda_1 + 1\lambda_2 + 1\lambda_1\lambda_2)\)
\(=> K ={1\over \sqrt{5}} (\lambda_1+1 -\lambda_1\lambda_2 - \lambda^2_2 + \lambda_2 +1\lambda^2_1 -\lambda_2 -1 -\lambda _2 -1 + \lambda_1\lambda_2)\)
\(=> K ={1\over \sqrt{5}}\bigg({\lambda_1 + 1 - {1+\sqrt{5}\over 2}}.{1-\sqrt{5}\over 2}0\ 0- \lambda_2 - 1 + {1+\sqrt{5}\over 2}.{1-\sqrt{5}\over 2}\bigg)\)
Since, λ^{2 }- λ - 1 = 0
\(=> K ={1\over \sqrt{5}} (\lambda_1+ 1 + 1\ 0\ 0 - \lambda_2- 1 -1)\)
\(=> K ={1\over \sqrt{5}}\biggl({1+\sqrt{5}\over 2}+ 2\ 0\ 0 \ - {1-\sqrt{5}\over 2} -2\biggl)\)
\(=> K ={\sqrt{5}\over 5}\biggl({5+\sqrt{5}\over 2}\ 0\ 0 \ {-5 + \sqrt{5}\over 2} \biggl)\)
\(=> K =\biggl({5\sqrt{5}+5\over 10}\ 0\ 0 \ {- 5\sqrt{5} + 5\over 10} \biggl)\)
\(=> K =\biggl({\sqrt{5}+1\over 2} \ 0\ 0 \ {- \sqrt{5} + 1\over 2} \biggl)\)
\(=> K =(\lambda _1\ 0\ 0\ \lambda_2)\)
Now, we know that
X = M . K . M^{-1}
^{\(=>X^{n-1}\ = ( M.K.M^{-1})^{n-1}\)}
=> X^{n-1 }= M . K . M^{-1 }M . K . M^{-1 }M . K . M^{-1 }M . K . M^{-1 }…………(upto n^{-}^{1} times) M . K . M^{-1}
=> X^{n-1 }= M . K . K . K . K …………(upto n^{-1} times) K . M^{-1}
=> X^{n-1 }= M . K^{n-1}. M^{-1}
\(∴ K^{n-1} = \biggl(\lambda _1^{n-1}\ 0\ 0 \ \lambda_2^{n-1}\biggl)\)
Initially we represented the nth term in terms of first two terms as:
\(\biggl({T_n\over T_{n-1}}\biggl)= X^{n-1}\biggl({T_1\over T_0}\biggl) = X^{n-1}\biggl({1\over 0}\biggl)\)
\(\biggl({T_n\over T_{n-1}}\biggl)= M.K^{n-1}.M^{-1} \biggl({1\over 0}\biggl)\)
\(=> \biggl({T_n\over T_{n-1}}\biggl) \ =(\lambda _1 \lambda_2\ 1\ 1 ).\biggl(\lambda_1^{n-1}\ 0 \ 0 \ \lambda _2^{n-1}\biggl) . {1\over \sqrt{5}}(1 - \lambda_2\ - \ 1\lambda_1)({1\over 0})\)
\(=> \biggl({T_n\over T_{n-1}}\biggl) \ = {1\over \sqrt{5}} (\lambda_1 \lambda_2\ 1\ 1).\biggl({\lambda_1^{n-1}\over -\lambda_2^{n-1}}\biggl)\)
\(=> \biggl({T_n\over T_{n-1}}\biggl) \ = {1\over \sqrt{5}}\biggl({\lambda_1^{n}- \lambda ^n_2\over -\lambda_1^{n-1}- \lambda_2^{n-1}}\biggl)\)
\( ∴ T_n= {1\over \sqrt{5}}(\lambda^n_1\ - \lambda^n_2)\)
\(=> T_n = {1\over \sqrt{5}}({1+\sqrt{5}\over 2})^n\ - ({1-\sqrt{5}\over 2})^n\)
\(∴ T_{n-1} = {1\over \sqrt{5}} (\lambda^{n-1}_1\ - \lambda_2^{n-1})\)
\(=> T_{n-1} = {1\over \sqrt{5}} ({1+\sqrt{5}\over 2})^{n-1} - ({1-\sqrt{5}\over 2})^{n-1}\)
Lastly, the limiting ratio of two consecutive terms is given below:
\(T_n\over T_{n-1}\)
\({1\over \sqrt{5}}(\lambda_1^n - \lambda_2^n)\over {1\over \sqrt{5}}(\lambda_1^{n-1}\ - \lambda_2^{n-1})\)
\(\lambda_1^n({1 {\lambda^n_2\over \lambda_1^n}})\over \lambda_1^{n-1}(1 - {\lambda_2^{n-1}\over \lambda_1^{n-1}}))\)
\(\lambda_1({1 - {\lambda^n_2\over \lambda_1^n}})\over (1 - {\lambda_2^{n-1}\over \lambda_1^{n-1}}))\)
\(({1 - {\lambda^n_2\over \lambda_1^n}})\over (1 - {\lambda_2^{n-1}\over \lambda_1^{n-1}}))\)
\(= \lambda _1 = {1+\sqrt{5}\over 2}\) = 1.61803399
\(\lambda_2\over \lambda_1\)
\(= {{1-\sqrt{5}\over 2}\over {1+ \sqrt{5}\over 2}}\)
\(= {{(1-\sqrt{5}})^2\over (1+ \sqrt{5})(1-\sqrt{5}) }\)
\(= {1-2\sqrt{5} + 5 \over 1-5}\)
\(= { 6-2\sqrt{5}\over -4}\)
\(= - { 3- \sqrt{5}\over 2}\)
≈ - 0.3819
\(- 1< {\lambda_2\over \lambda_1}<0\)
\(=> 0 < \biggl|{\lambda_2\over \lambda_1}\biggl|<1\)
\(=> ({\lambda_2\over \lambda_1})^n \ = 0\)
Thus, we can state that the ratio between any two term of a Fibonacci Sequence is 1.618. It should be noted that the ratio is constant for any two term of the Fibonacci Sequence because the Fibonacci function has been expressed as an Eigen Vector and is the Eigen Value of the vector. Eigen Value is the constant value for all the terms of the function. However, it has been verified in the next section.
Furthermore, this value is also equal to the golden ratio of the Fibonacci Spiral.
Moreover, this is the value of Eigen Value . Thus, it also verifies that v is the Eigen Vector of Matrix X. Furthermore, the straight line (2) is actually the Eigen Vector because all the points that lie on the Eigen Vector are actually the terms of the sequence and the next term can be found by simply multiplying the present term with the Eigen Value.
Case 1: We will find the next terms using the Golden Ratio 1.61803399. Let us assume u_{3 }= 1.
Term (u_{n})
Observed Value
u_{n }= u_{n-1}×1.618
1
1
2
2
1.6180
3
3
3.2360
4
5
4.8541
5
8
8.0901
6
13
12.9442
7
21
21.0344
8
34
33.9787
9
55
55.0131
10
89
88.9918
11
144
144.0050
12
233
232.9968
Sample Calculation:
Observed Value of 5^{th} Term = (Value of 4^{th} Term) × 1.618
= 2 × 1.618 = 3.236 ≈ 3
Error of 5^{th} Term = 3.2360 - 3 = 0.2360
The above graph shows the variation between absolute error in determining the value of any term of the Fibonacci sequence using Golden ratio with respect to the number of terms. It is observed from the graph that the absolute error decreases exponentially with respect to the number of terms. The value of absolute error decreases from 0.382 to 0.003 when the number of terms is increased from 2 to 12 at a regular interval of 1 term. The equation of trend obtained in the graph using MS Excel is shown below:
y = 1.005 × e^{-0.483x}
In the above-mentioned equation, y represents the absolute error of any term and x denotes the number of terms. Analyzing the above equation, it can be said that:
0 = 1.005 × e^{-0.483x}
=> e^{-0.483x }= 0
Taking logarithm in both sides of the equation:
ln ln e^{-0.483x} = ln ln 0
=> - 0.483x × ln ln e = - ∞
=> x = ∞
Hence, it can be said that the absolute error in determining the value of the term will be equal to zero for nth term where n = ∞.
Lastly, there are no significant outliers in the graph which strengthen the analysis and the equal of trend. It is again justified by the maximum value of Regression coefficient.
So, let as assume that, if n tends to infinity, the value of u_{n} will tend to become constant (u).
Mathematically, it can written as:
u_{n} = \(\frac{1}{1+u_{n-1}}\){for n >1}
=> u_{n} + u_{n }u_{n - 1 }= 1
Taking limits in both sides:
u_{n }+ u_{n} u_{n - 1} = 1
=> u_{n }+ u_{n} u_{n - 1} = 1
=> u_{n }+ u_{n} u_{n - 1} = 1
=> u_{ }+ u.u = 1
=> u^{2} + u - 1 = 0
Solving the above equation using Sreedhar Acharya’s Formula, we get:
\(u=\frac{-1±\sqrt{1+4}}{2}\)
\(=>u=\frac{-1±\sqrt{5}}{2}\)
From the above study of the sequence, we can conclude that, there cannot be any negative value of terms of the sequence as n >1. Therefore, we can write,
\(u=\frac{-1+\sqrt5}{2}\)
=> u = 0.618
Case 2: We will find the previous terms using the Golden Ratio, 0. 61803399. Let us assume u_{14 }= 233.
Term (u_{n})
Observed Value
u_{n }= u_{n }_{-1 }× 0.618
Sample Calculation
Observed Value of 11^{th} Term = (Value of 12^{th} Term) × 0.618
= 89 × 0.618 = 55.0050 ≈ 55
Error of 11^{th} Term = 55.0050 - 55 = 0.0050
The above graph shows the variation between absolute error in determining the value of any term of the Fibonacci sequence using Golden ratio with respect to the number of terms. It is observed from the graph that the absolute error decreases exponentially with respect to the number of terms. The value of absolute error decreases from 0.236 to 0.001 when the number of terms is increased from 1 to 11 at a regular interval of 1 term. The equation of trend obtained in the graph using MS Excel is shown below:
y = 0.3824 × e^{-0.482x}
In the above-mentioned equation, y represents the absolute error of any term and x denotes the number of terms. Analyzing the above equation, it can be said that:
0 = 0.3824 × e^{-0.482x}
=> e ^{-0.482x }= 0
Taking logarithm in both sides of the equation:
ln ln e^{-0.482x} = ln ln 0
=> -0.482x × ln ln e = - ∞
=> x = ∞
Hence, it can be said that the absolute error in determining the value of the term will be equal to zero for nth term where n = ∞.
Lastly, there are no significant outliers in the graph which strengthen the analysis and the equal of trend. It is again justified by the maximum value of Regression coefficient.
There exists an array of applications of Fibonacci Sequence in the real world. A few of the most important applications are as follows:
What is the generalized scaling factor which should be multiplied with any term of the Fibonacci Sequence to get the next term?
In this exploration, two scaling factors has been derived to find the next term of the Fibonacci Sequence. Firstly, if the progression is done in ascending order, then a factor of 1.618 should be multiplied with nth term to get (n + 1) th term. Secondly, if the progression is done in descending order, then a factor of 0.618 should be multiplied with nth term to get (n - 1)th term.
\(T_{n-1}=\frac{1}{\sqrt5}(\frac{1+\sqrt5}{2})^{n-1}-(\frac{1-\sqrt5}{2})^{n-1}\)
In this investigation, several process and mathematical tools have been observed to derive the nth term of a Fibonacci Sequence. Two ways are used, one using matrix analysis and eigen vector and another one using a function to derive the nth term of a Fibonacci Sequence. As both of the procedures has led to same results, it increases the existence of mathematical calculation and exploration of this IA. Both graphical and tabular methodology has been observed to verify the existence of the scaling factors obtained, eigen values etc. Lastly, the error analysis has also been a strength of the IA.
However, there are few weakness that has been observed during this mathematical investigation. As the function of a nth term of a Fibonacci Sequence solely depends upon the previous term and has no correlation with respect to any other term, the process that we have observed is not error free. However, the error has been minimised by taking least approximation possible.
Golden ratio is one of the mostly observed mathematical value which is observed in many historic monuments, architecture, human anatomy and many more. Though in this exploration, the value of golden ratio was determined using Fibonacci Sequence; however, there are various methods and approaches to find the value of the golden ratio from scratch. It is obtained in this exploration that determination of golden ratio using an Fibonacci Sequence or Fibonacci Spiral gives an error during its application, i.e., the value of the terms obtained incurred some error. It is again studied that the error in infinite-th term will be zero. This makes the value of golden ratio questionable. Though the obtained value of golden ratio in this exploration is very close to the accurate value, however, as the value of the terms of Fibonacci sequence obtained are not free from errors, there must be some error in the value of golden ratio. Thus, another exploration should be framed to determine the value of golden ratio using any other method and a generic comparison should be established between the approach to be followed and the approach that is followed in this exploration. The research question of the exploration is stated as: “Investigation on the derivation and hence determination of golden ratio using two methods – “Fibonacci Sequence using Eigen Vector”, and “Method of Pentagram” and comparison between the two based on the error percentage of the obtained value of golden ratio?