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.

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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.

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I rushed to my father to tell him about it. &nbsp;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.

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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.

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In the process of my study, I came across the term, &ldquo;golden ratio.&rdquo; 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&rsquo;s 5th symphony fit the golden ratio; if a musical note maintains the ratio, it becomes soothing to the ear.

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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 &nbsp;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:

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Tn&nbsp;= Tn&nbsp;- 1&nbsp;+ Tn&nbsp;- 2&nbsp;&nbsp;..............(equation - 1)

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Example of a Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 &hellip;

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 (Tn) and ordinate of each point as the previous term (Tn -&nbsp;1). The graphs have been plotted using online tool Desmos.

fibonacci-series.svg

Series.svg

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:

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y = 0.6181x - 0.0330268&nbsp;&hellip;&hellip;&hellip;&hellip;&hellip;(equation -&nbsp;2)

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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.

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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 &times; 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.

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Then the matrix will be:

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\(\biggl({T_n\over T_{n-1}}\biggl)\ =(1\ 1\ 1\ 0)\biggl({T_{n-1}\over T_{n-2}}\biggl)\)&hellip;&hellip;&hellip;&hellip;&hellip;(equation -&nbsp;3)

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\(=&gt; \biggl({x\over y}\biggl)=(1\ 1\ 1\ 0)\biggl({T_{n-1}\over T_{n-2}}\biggl) \).&hellip;&hellip;&hellip;&hellip;&hellip;(equation -&nbsp;4)

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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:

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\(u_1 = 1,u_n = {1\over 1+u_{n-1}}\{for n &gt; 1\}\)

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The first five terms of this sequence are given as follows:

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u1&nbsp;= 1

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\(u_2 =  {1\over 1+u_1} \ ={1\over 1+1} = 0.5 \)

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\(u_3 {1\over 1+u_2} \ ={1\over 1+0.5} = 0.6667\)

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\(u_4 ={1\over 1+u_3} \ ={1\over 1+0.6667} = 0.6000\)

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\(u_5 = {1\over 1+u_4} \ ={1\over 1+0.6000} = 0.625\)

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Value of all the subsequent terms are solved using a python code which is shown. The python code was simulated in PyCharm IDE.

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From the above result, we can see that, 11th term (u11) onwards, the value of every term is approximately equal to 0.618.

representation.svg

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.

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The Eigen Vector \(\vec{ v}\) of any Matrix A may be represented as the equation stated below, where \(\lambda\)&nbsp;is the Eigen Value of the Eigen Vector.

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\(\)

\(A \vec{ v}\ =\ \lambda \vec{ v}\)

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\(=&gt; A \vec{ v}\ =\ \lambda \vec{ v}\)&hellip;&hellip;&hellip;&hellip;&hellip;&hellip;&hellip;&hellip;(equation - 6)

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Here I is an identity matrix with the order of Matrix A.

Eigen Value &nbsp;of an Eigen Vector may be defined as the factor by which it is scaled.

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From Equation (6), let us derive the formula of Eigen Value:

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\(A\vec v \ = \lambda I \ \vec v\)

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\( =&gt; A\vec v \ = \lambda I \ \vec v = 0\)

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\( =&gt; [A \ - \lambda I] \ \vec v = 0\)&nbsp;&hellip;&hellip;&hellip;&hellip;&hellip;&hellip;&hellip;&hellip;&hellip;(equation - 7)

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From the definition of Eigen Vector, \(\vec v\) is a non-zero vector. So, we can write:

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det det (&nbsp;A - &lambda;I&nbsp;) = 0 &hellip;&hellip;&hellip;&hellip;&hellip;&hellip;&hellip;&hellip;&hellip;&hellip;(equation - 8)

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This is defined as the Characteristic Polynomial of a matrix.

Case 1&nbsp;We will find the next terms using the Golden Ratio 1.61803399. Let us assume u3&nbsp;= 1.

Sample Calculation

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Observed Value of 5th Term = (Value of 4th Term) &times; 1.618

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= 2 &times; 1.618 = 3.236&nbsp; &asymp; 3

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Error of 5th Term = 3.2360 - 3 = 0.2360

Group 7.svg

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:

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y = 1.005 &times; e-0.483x

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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:

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0 = 1.005 &times; e-0.483x

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=&gt; e-0.483x&nbsp;= 0

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Taking logarithm in both sides of the equation:

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ln ln e-0.483x = ln ln 0

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=&gt; - 0.483x &times; ln ln e = - &infin;

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=&gt; x = &infin;

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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 = &infin;.

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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 un will tend to become constant (u).

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Mathematically, it can written as:

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\(u_n = \frac{1}{1+u_{n-1}}\{for n &gt;1\}\)

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=&gt; un + un&nbsp;un&nbsp;- 1&nbsp;= 1

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Taking limits in both sides:

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un&nbsp;+&nbsp;un un&nbsp;- 1 = 1

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=&gt;&nbsp;un&nbsp;+&nbsp;un un&nbsp;- 1 = 1

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=&gt;&nbsp;un&nbsp;+&nbsp;un un&nbsp;- 1 = 1

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=&gt;&nbsp;u&nbsp;+&nbsp;u.u&nbsp;= 1

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=&gt; u2 + u - 1 = 0

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Solving the above equation using Sreedhar Acharya&rsquo;s Formula, we get:

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\(u=\frac{-1±\sqrt{1+4}}{2}\)

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\(=&gt;u=\frac{-1±\sqrt{5}}{2}\)

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From the above study of the sequence, we can conclude that, there cannot be any negative value of terms of the sequence as n &gt;1. Therefore, we can write,

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\(u=\frac{-1+\sqrt5}{2}\)

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=&gt; u = 0.618

Case 2: We will find the previous terms using the Golden Ratio, 0. 61803399. Let us assume u14&nbsp;= 233.

Derivation Of Nth Term Of Fibonacci Series Using The First Two Terms

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Ilan Grapel

Tech & Law Grad Educator: Boost IB Math/CS Skills with Real-World Projects & Expertise!