...FIBONACCI SEQUENCE Shafira Chairunnisa | 11 Blue INTRODUCTION Before we get into the details of Fibonacci sequence, let us go back to the basic definition of a sequence in Mathematics. According to The Free Dictionary by Farlex, a sequence is an ordered set of mathematical quantities called terms. There are two types of sequence: arithmetic and geometric sequence. An arithmetic sequence is a sequence of numbers that has a constant difference between every two consecutive terms. 1, 3, 5, 7, 9,…. The numbers in the sequence are called terms. Thus, 1 is the first term, 3 is the second term, 5 is the third term, and so forth. The symbol Un denotes the first term of a sequence. Since the first term is 1, we can express it as Un= 1. Each consecutive terms has a difference (denoted as d). Meanwhile, a geometric sequence is a sequence in which the same number is multiplied or divided by each term to get the next term in the sequence. 3, 9, 27, 81, 243,…. This is an example of geometric sequence. The quotient of a term with its previous term is called ratio. In geometric sequence, the ratio between each successive terms is constant. What is a Fibonacci sequence? Fibonacci sequence is a special sequence. It is founded by an Italian mathematician named Leonardo Pisano Bigollo, known commonly as Fibonacci. According to WhatIs.com, it is a set of numbers that starts with a one or a zero, followed by a one, and proceeds based on the rule that each number (called...
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...Arithmetic and geometric progressions mcTY-apgp-2009-1 This unit introduces sequences and series, and gives some simple examples of each. It also explores particular types of sequence known as arithmetic progressions (APs) and geometric progressions (GPs), and the corresponding series. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. After reading this text, and/or viewing the video tutorial on this topic, you should be able to: • recognise the difference between a sequence and a series; • recognise an arithmetic progression; • find the n-th term of an arithmetic progression; • find the sum of an arithmetic series; • recognise a geometric progression; • find the n-th term of a geometric progression; • find the sum of a geometric series; • find the sum to infinity of a geometric series with common ratio |r| < 1. Contents 1. Sequences 2. Series 3. Arithmetic progressions 4. The sum of an arithmetic series 5. Geometric progressions 6. The sum of a geometric series 7. Convergence of geometric series www.mathcentre.ac.uk 1 c mathcentre 2009 2 3 4 5 8 9 12 1. Sequences What is a sequence? It is a set of numbers which are written in some particular order. For example, take the numbers 1, 3, 5, 7, 9, . . . . Here, we seem to have a rule. We have a sequence of odd numbers. To put this another way, we start with the number 1, which is an odd number, and then each successive number is obtained...
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...Brofft 23 May 2012 Fibonacci's Sequence The Fibonacci Sequence is one of the most famous, if not the most famous numerical sequence in history. "In the year 1202, mathematician Leonardo of Pisa (also known as Fibonacci) posed the following problem: How many pairs of rabbits will be produced in a year, beginning with a single pair, if every month each pair bears a new pair that becomes productive from the second month on? The total number of pairs, month by month, forms the sequence 1,1,2,3,5,8,13,21,34,55,89, and so on. Each new term is the sum of the previous two terms." (1) As mentioned previously, "the solution of this problem leads to a sequence of numbers known as the Fibonacci sequence. Here are the first fifteen terms of the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 (equivalent to 1.618). After the first two terms, in the sequence, each term is obtained by adding the two previous terms. For example, the third term is obtained by adding 1 + 1 to get 2, the fourth term is obtained by adding 1 + 2 to get 3, and so on." (2) Leonardo of Pisa, born in 1170, grew up and traveled throughout Africa and the Mediterranean during much of his childhood and early adult life, learning different mathematical styles and formulas, which were of great interest to him. During these travels, Fibonacci (nicknamed many years after his death; 'Bonacci' means "son of good fortune". While living, Fibonacci went by Leonardo of Pisa, his hometown...
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...Fibonacci Numbers In the 13th century a man named Leonardo of Pisa or Fibonacci founded Fibonacci Numbers. Fibonacci Numbers are “a series of numbers in which each number is the sum of the two preceding numbers” (Burger 57). His book “Liber Abaci” written in 1202 introduced this sequence to Western European mathematics, although they had been described earlier in Indian mathematics. He proved that through spiral counts there is a sequence of numbers with a definite pattern. The simplest series is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…and so on. When looking at this series the pattern proves that adding the previous number to the next will give you the following number in the series. For example, (1+1=2), (2+3=5), and etc. In order to ensure accuracy when using Fibonacci Numbers a formula was created. The formula or rule that follows the Fibonacci sequence is Fn = Fn-1 + Fn-2. By plugging in any numbers in a problem to this equation a student can find the right answer. This gives students the ability to calculate any Fibonacci Number. In modern times society uses these numbers to calculate numerous things. For instance, like the sizes of our arms relative to our torso and even the structure of hurricanes. On another note, Fibonacci Numbers can also be found in patterns in nature. It is truly astonishing to think about how relations in Fibonacci Numbers may possibly be represented in our lives. Works Cited Burger, Edward B., and Michael P. Starbird....
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...Pamela Dela Cruz Math 150A M/Tu/W/F 10-10:50am. 12/10/12 Leonardo Piscano Fibonacci In 1170, Leonardo Piscano Fibonacci, more commonly referred to as Fibonacci, was born in Pisa, Italy to Guilielmo, a member of the Bonacci family. Guilielmo held a position as a secretary of the Republic of Pisa, in the province of Tuscany. In 1192, Guilielmo was posted to trading center in the city of Bugia in North Africa. After accepting this job, Guiliemo brought Fibonacci with him1. Here, Fibonacci was taught his education. He continued to learn more as he grew motivation to mathematical inquiry from traveling around other countries with his father2. Somewhere in between Barbary and Constantinople, he had become familiar with the Hindu/Arabic numeral system and discovered its enormous practical advantages compared to the Roman Numerals. He had ended his travels in 1200 and returned back to Pisa. For the next 25 years, he wrote a number of texts that played an important role in reviving ancient mathematical skills or contributed to the making of his own1. Four of which became the major components to his surviving work. In 1202, he released his first major work, “Liber Abbaci” or “The Book of Calculations”. It had presented a basic overview of basic arithmetic and algebra. However, he had also exposed a new alternative computing method for the replacement of having to use an abacus for arithmetical operations and was based on written algorithms rather counting objects. First, he discussed...
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...Our Project on the European Number System Our Project on the European Number System Names Teacher Grade/Form Renee Gordon Mrs.Mcneil 71 Janice Ranns Javon Wright Tommy-Lee Lindo Table of Contents Introduction…………………………………………………………………. (i) History of European Mathematics…………………………………………….(1) Famous European Mathematician………………………………...(2)-(3) Reference Page………………………………………………………………………(4) Introduction (i) This project introduces you to, the European’s Contributions to the Number system/Numerical system. This project also shows, you famous mathematicians, and more so sit back, read and enjoy this project. (1) History of European Mathematics During the centuries in which the Chinese, Indian and Islamic mathematicians had been in the ascendancy, Europe had fallen into the Dark Ages, in which science, mathematics and almost all intellectual endeavour stagnated. Scholastic scholars only valued studies in the humanities, such as philosophy and literature, and spent much of their energies quarrelling over subtle subjects in metaphysics and theology. From the 4th to 12th Centuries, European knowledge and study of arithmetic, geometry, astronomy and music was limited mainly to Boethius’ translations of some of the works of ancient Greek masters such as Nicomachus and Euclid. All trade and calculation was made using the clumsy and inefficient Roman numeral...
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...The Influence of Mathematics on Daily Social Activities MTH/110 March 2, 2015 The Influence of Mathematics Society is an important concept in which stands the intellectual development of mathematics and science. Today, in society, mathematics makes extraordinary demands and so to the world of mathematicians and scientists. The purpose of the report is to identify the influence and effects of mathematics in society. It includes some history of math and its major demands and capabilities that come with the material and mathematicians. Math Background People have believed in mathematics and the discipline that comes with it since centuries ago, some may like, and some may not. Some people used to worship mathematics and believe in it for living like Aryabatta and Bhaskara. Back to the 4th century, Aristotle and Plato had already an idea of the existence of mathematics in their mind and the external world; also he argued about a positive effect on individuals (Dossey). In the middle ages, mathematicians were coming out and at one point competing without knowing in discovering new techniques. Archimedes had one of the greatest impacts on its work in mathematics but he was known later in the 16th century when Federico Commandino in 1558 translation into Latin most of his printing texts and spread it out with other mathematicians and physics of the time, that includes Johannes Kepler and Galileo Galilei (Toomer, 2014). Many other mathematicians...
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...Leonardo Pisano Bigollo was an Italian Mathematician who was born in 1170 in Pisa, Italy. His parents were Alessandra Bonacci and Guglielmo Bonacci he was educated in North Africa when he was younger he grew up loving math. His parents sent him away to go study calculation with an Arab Master (O'Connor, JJ, and EF Robertson). Fibonacci came up with the the Golden Ratio which is a number that dividing a line in two parts so therefore the longer part is equal to the smaller part which is equal to the whole length divided. He was an great man that had a childhood but everyone else figured he did not because no one ever knew of anything. Nobody ever knew of Leonardo because it’s like his parents kept him a secret and hid him from the world until...
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...Vineet Iyer Process Decryption Fibonacci. 1. The Mathematics Mathematicians, scientists and naturalists have known this ratio for years. It's derived from something known as the Fibonacci sequence, named after its Italian founder, Leonardo Fibonacci (whose birth is assumed to be around 1175 AD and death around 1250 AD). Each term in this sequence is simply the sum of the two preceding terms (1, 1, 2, 3, 5, 8, 13, etc.). But this sequence is not all that important; rather, it is the ratio of the adjacent terms that possess great qualities, roughly 1.618, or its inverse 0.618. This proportion is known by many names: the golden ratio, the golden mean, PHI and the divine proportion, among others. Almost everything that has dimensional properties adheres to the ratio of 1.618. 2. Prove It! Take honeybees, for example. If you divide the female bees by the male bees in any given hive, you will get 1.618. Sunflowers, which have opposing spirals of seeds, have a 1.618 ratio between the diameters of each rotation. This same ratio can be seen in relationships between different things throughout nature. Still don't believe it? Try measuring from your shoulder to your fingertips, and then divide this number by the length from your elbow to your fingertips. Or try measuring from your head to your feet, and divide that by the length from your belly button to your feet. The golden ratio is seemingly unavoidable. The markets have the very same mathematical base as these natural...
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...Maths in nature "The laws of nature are but the mathematical thoughts of God" - Euclid Mathematics is everywhere in this universe. We seldom note it. We enjoy nature and are not interested in going deep about what mathematical idea is in it. Here are a very few properties of mathematics that are depicted in nature. SYMMETRY Symmetry is everywhere you look in nature . Symmetry is when a figure has two sides that are mirror images of one another. It would then be possible to draw a line through a picture of the object and along either side the image would look exactly the same. This line would be called a line of symmetry. There are two kinds of symmetry. One is bilateral symmetry in which an object has two sides that are mirror images of each other. The human body would be an excellent example of a living being that has bilateral symmetry. The other kind of symmetry is radial symmetry. This is where there is a center point and numerous lines of symmetry could be drawn. The most obvious geometric example would be a circle. Shapes Sphere: A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. The shape of the Earth is very close to that of an oblate spheroid, a sphere flattened along the axis from pole to pole such that there is a bulge around the equator. The wee electron has gotten its most thorough physical examination yet, and scientists report that it is almost, almost a perfect...
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...The Da Vinci Code, which was released in the May of 2006, was about a murder that took place in the Louvre in France. There were clues in all of Da Vinci paintings throughout the museum. These clues lead to a discovery of a religious mystery that has been protected and kept hidden for two thousand years. This throws Robert Langdon and the victim’s granddaughter into a bizarre murder and crazy mystery, Not only is this movie a good movie and an interesting mystery to watch. But I can also connect what I’ve learned in sacred geometry to this movie in many aspects. One of the first examples of something I connected to our class, sacred geometry, was the shape of the museum that the dead body was found. The building that the body was found in was the Louvre museum. The Louvre museum is located in Paris, France and was established in 1793. In front of the actual museum there is something that is known as the Louvre pyramid. The Louvre pyramid is a large glass and metal pyramid surrounded by three smaller pyramids. The large pyramid serves as the entrance to the Louvre museum and was opened in 1989. The reason that I can connect this to our course in sacred geometry is the actually structure of the pyramid. There are many small triangles on it, which are also known as the triad. The triad is a three-sided shape and also is known as the first and oldest number. Also this is a pyramid and has many pyramids throughout it. The pyramid is something that we talked about in class to as...
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...Write the first four terms of the sequence whose general term is an = 2( 4n - 1) (Points : 3) | 6, 14, 22, 30 -2, 6, 14, 22 3, 7, 11, 15 6, 12, 18, 24 | 2. Write the first four terms of the sequence an = 3 an-1+1 for n ≥2, where a1=5 (Points : 3) | 5, 15, 45, 135 5, 16, 49, 148 5, 16, 46, 136 5, 14, 41, 122 | 3. Write a formula for the general term (the nth term) of the arithmetic sequence 13, 6, -1, -8, . . .. Then find the 20th term. (Points : 3) | an = -7n+20; a20 = -120 an = -6n+20; a20 = -100 an = -7n+20; a20 = -140 an = -6n+20; a20 = -100 | 4. Construct a series using the following notation: (Points : 3) | 6 + 10 + 14 + 18 -3 + 0 + 3 + 6 1 + 5 + 9 + 13 9 + 13 + 17 + 21 | 5. Evaluate the sum: (Points : 3) | 7 16 23 40 | 6. Find the 16th term of the arithmetic sequence 4, 8, 12, .... (Points : 3) | -48 56 60 64 | 7. Identify the expression for the following summation:(Points : 3) | 6 3 k 4k - 3 | 8. A man earned $2500 the first year he worked. If he received a raise of $600 at the end of each year, what was his salary during the 10th year? (Points : 3) | $7900 $7300 $8500 $6700 | 9. Find the common ratio for the geometric sequence.: 8, 4, 2, 1, 1/2 (Points : 3)...
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...How satisfied are you with Wikipedia? Your feedback is important to us! As a token of appreciation for your support you get a chance of winning a Wikipedia T-shirt. Click here to learn more! Calculus From Wikipedia, the free encyclopedia Jump to: navigation, search This article is about the branch of mathematics. For other uses, see Calculus (disambiguation). | It has been suggested that Infinitesimal calculus be merged into this article or section. (Discuss) Proposed since May 2011. | Topics in Calculus | Fundamental theorem Limits of functions Continuity Mean value theorem [show]Differential calculus | Derivative Change of variables Implicit differentiation Taylor's theorem Related rates Rules and identities:Power rule, Product rule, Quotient rule, Chain rule | [show]Integral calculus | IntegralLists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution, partial fractions, changing order | [show]Vector calculus | Gradient Divergence Curl Laplacian Gradient theorem Green's theorem Stokes' theorem Divergence theorem | [show]Multivariable calculus | Matrix calculus Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian | | Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes...
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...MCR 3U Exam Review Unit 1 1. Evaluate each of the following. a) b) c) 2. Simplify. Express each answer with positive exponents. a) b) c) 3. Simplify and state restrictions a) b) c) d) 4. Is Justify your response. 5. Is Justify your response. Unit 2 1. Simplify each of the following. a) b) c) 2. Solve. a) b) c) 3. Solve. Express solutions in simplest radical form. a) b) 4. Find the maximum or minimum value of the function and the value of x when it occurs. a) b) 5. Write a quadratic equation, in standard form, with the roots a) and and that passes through the point (3, 1). b) and and that passes through the point (-1, 4). 6. The sum of two numbers is 20. What is the least possible sum of their squares? 7. Two numbers have a sum of 22 and their product is 103. What are the numbers ,in simplest radical form. Unit 3 1. Determine which of the following equations represent functions. Explain. Include a graph. a) b) c) d) 2. State the domain and range for each relation in question 1. 3. If and , determine the following: a) b) 4. Let . Determine the values of x for which a) b) Recall the base graphs. 5. Graph . State the domain and range. Describe...
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...Riphah International University, Faisalabad Visual Programming – Assignment 2 Functions and Control Sructures Instructions for Submission Submission Guidelines: • Your code must be properly commented and your code should be neat and nested format. • All the steps involved in solution of each question should be written. Just Answers are not required • Try NOT to copy paste data from your friends etc. • This is an individual assignment. PLAGIARISM IS NOT ACCEPTABLE! In case of plagiarism you will get ZERO MARKS for that question • Please do not copy paste program from your friend. • Complete the assignment nicely and submit .cpp files on email you can zip all program in one file then submit that file • Submit files in format like “yourname_yourclass_q1.cpp, yourname_yourclass_q2.cpp”, other file naming convention will not be accepted. • Submit files on uzairsaeed@riphahfsd.edu.pk • Other questions except program can be solved on Microsoft Word or alike, and plagiarism is 30% with proper reference is allowed only • Last date of submission 02-October-2013 11:55 pm Question 1: Write a function that receives 5 integers and returns the sum and average of these numbers. Call this function from main( ) and print the results in main(). Question 2: A 5-digit positive integer is entered through the keyboard, write a function to calculate sum of digits of the 5-digit number. Question 3: A positive integer is entered through the keyboard and taking input and terminates on sentinel value...
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