The Fibonacci Sequence and Generalizations Essay Example

The Fibonacci Sequence and Generalizations Paper Abstract: This paper gives a brief introduction to the famous Fibonacci sequence and demonstrates the close link between matrices and Fibonacci numbers. The much-studied Fibonacci sequence is defined recursively by the equation yk+2 = yk+1 + yk, where y1 = 1 and y2=1. By using algebraic properties of matrices, we derive an explicit formula for the kth Fibonacci number as a function of k and an approximation for the golden ratio yk+1 / yk. We also demonstrate how useful eigenvectors and eigenvalues can be in understanding the dynamics of linear recurrence relations of the form yk+2 = ayk+1 + byk where a, b R. I. Introduction  The Fibonacci sequence, probably one of the oldest and most famous sequences of integers, has fascinated both amateur and professional mathematicians for centuries. Named after its originator, Leonardo Fibonacci, the Fibonacci sequence occurs frequently in nature and has numerous applications in applied and pure mathematics.  The Fibonacci sequence is the sequence of numbers:  0, 1, 1, 2, 3, 5, 8, 13, 21,  where each member of the sequence is the sum of the preceding two. Therefore, the nth Fibonacci number is defined recursively as follows:  y1 = y2 = 1 (1)  yn = yn-1 + yn-2 n 3  Historically, this sequence appeared for the first time in a problem posed by the Italian scholar Leonardo Fibonacci in 1202. In his famous work Liber Abaci, Leonardo Fibonacci asked the following famous question on the rate growth of rabbits: We will write a custom essay sample on The Fibonacci Sequence and Generalizations specifically for you for only $16.38 $13.9/page Order now We will write a custom essay sample on The Fibonacci Sequence and Generalizations specifically for you FOR ONLY $16.38 $13.9/page Hire Writer We will write a custom essay sample on The Fibonacci Sequence and Generalizations specifically for you FOR ONLY $16.38 $13.9/page Hire Writer Suppose that on January, 1st there are two newborn rabbits, one male and one female. What is the number of rabbits produced in a year if the following conditions hold:  1) each pair takes one month to reach maturity  2) each pair produces a mixed pair of rabbits every month, from February on; and  3) no rabbits die during the course of the year.  Assume that on January 1st there is one mixed pair of baby rabbits. At the beginning of February there will still be one pair of rabbits since it takes a month for them to become mature and reproduce. In February one mixed pair of rabbits will be produced and in March two pairs will be produced, one by the original pair and one by the pair produced in February. Following this pattern, in April, three pairs will be produced, and in May five pairs. Since the 22 matrix A has only one distinct eigenvalue with multiplicity 2, it follows that A is not diagonalizable. Therefore, there is no general formula for Ak and hence for yk. From the two examples above we can derive the following two conclusions about our matrix method for solving second order recurrence relations:  1) If the characteristic equation of A has two distinct solutions, then A is diagonalizable and we can derive an explicit formula for yk.  2) If the characteristic equation of A has less than two distinct solutions, then A is not diagonalizable and our method for deriving yk cannot be applied.  More formally, our conclusions are consistent with the following more general theorem: Theorem: Let be the distinct solutions of the equation, where and b = 0. Then every solution of the linear homogenous recurrence relation with constant coefficients, where a0 = C0 and a1 = C1, is of the form  for some constants A and B.1  IV. Conclusion  This paper discussed how linear algebra can be applied to the analysis of linear recurrence relations, a number of which arise frequently in applied and pure math. In particular, it emphasized the concepts of eigenvalues and eigenvectors and how helpful they are in describing and understanding infinite sequences of integers. Bibliography: 1 For the proof of this theorem see Koshy, p.143-4

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Argumentative Essay Topics - Arguing With the South Korean Government

Argumentative Essay Topics - Arguing With the South Korean GovernmentThe South Korean government is being sued by the American Institute in South Korea (AISK) and several of its citizens for their involvement in a propaganda campaign to create the perception that the Japanese were not responsible for the sinking of the country's naval vessel, the Cheonan. The lawsuit says that this propaganda is causing severe psychological harm to its citizens and has become a justification for many self-aggrandizing actions against the country. However, the article you are reading provides an overview of the AISK complaint against the South Korean government, and the problems it presents for many key legal arguments that a group of American AISK members are making.One of the AISK members who have joined the lawsuit on behalf of a group of Korean citizens is Elliot E. Braithwaite, who founded the AISK Action Group to act as a watchdog to make sure the country does not violate international laws, par ticularly the United Nations Convention on the Law of the Sea (UNCLOS). He also served as the founding chairman of the AISK International Business Center, which is headquartered in San Francisco.On March 9, 2020, the AISK told the South Korean government that they had studied the Japanese Justice Ministry's investigations of the sinking of the Japanese fishing boat and determined that there were no damages to the cargo. In addition, they also showed that there were no items missing from the Cheonan that would have caused the collision.The plaintiffs in the case say that the South Korean government was indirectly responsible for the sinking, by failing to prevent it. They argue that the AISK Action Group has been very active in advocating that the South Korean government should join the International Court of Justice, which holds hearings on disputes between nations. According to these claims, South Korea will only join the court if they believe they can win an appeal.On April 8, 202 0, the South Korean government filed an appeal with the International Court of Justice. They argued that the court should decide the matter in accordance with UNCLOS. However, the plaintiffs argue that the Japanese, while a sovereign nation, are not entitled to be part of the court because they failed to comply with international law.According to the plaintiffs, there is no guarantee that Japan will abide by the court's rulings in this case. Furthermore, they contend that the argument that the action group's activities had been peaceful and that they were acting in the name of peace and justice is an unfair excuse given to justify their self-aggrandizing actions. Furthermore, the plaintiffs argue that the AISK Action Group's actions caused great damage to the country, and that any action based on this case will be viewed with greater suspicion by South Koreans.The plaintiffs argue that the government and their supporters have tried to divert attention away from their self-serving ac tions and focus on their debate support. However, the motion states that a significant portion of the evidence is irrelevant, irrelevant to the court's decision, and unreliable, rendering the case weak.The suit also states that they cannot fight against themselves or their own interpretation of the facts and need an expert who will make a more thorough analysis. There are so many different legal arguments that can be made on both sides of the issue, that it is best to stay ahead of the curve.

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