Here are the steps to solve this problem:
1) The first three terms of the Fibonacci sequence are given: 1, 1, 2
2) Use the recursive definition: Fn = Fn-1 + Fn-2 to generate more terms of the sequence until a term is greater than 500.
3) F4 = F3 + F2 = 2 + 1 = 3
4) F5 = F4 + F3 = 3 + 2 = 5
5) F6 = F5 + F4 = 5 + 3 = 8
6) F7 = F6 + F5 = 8 + 5 = 13
7) F8 = F7 + F6 = 13 + 8 = 21
This powerpoint presentation discusses about the first lesson in Grade 10 Math. It is all about Number Pattern. It also shows the definition, examples and how to find the nth term and general formula.
"Harmonic and Other Sequences" presentation includes a brief historical background, problems and solutions to the simplest problems which you may face in your Mathematics.
This powerpoint presentation discusses about the first lesson in Grade 10 Math. It is all about Number Pattern. It also shows the definition, examples and how to find the nth term and general formula.
"Harmonic and Other Sequences" presentation includes a brief historical background, problems and solutions to the simplest problems which you may face in your Mathematics.
Stress your brain to search the innumerable patterns hidden in numbers which we usually overlook in Maths.From Sierpinski Triangle to Pandiagonal Magic Square, Fibonnaci Numbers to Hockey Stick Pattern.Explore how they are used in daily life, in espionage.....?? technology and glare at the inventories.
Correlation of Fibonacci Sequence and Golden Ratio With its Applications in E...Dr. Amarjeet Singh
We have discussed in this elucidation paper about correlation of Fibonacci sequence and golden ratio with its applications in engineering and science. One of the most recurring sequences in nature is the Fibonacci sequence. As the sequence was explored, it was found out that this sequence led to the golden ratio. This study tried to apply the concept of Fibonacci and golden ratio to maximize efficiency of our live life. We consider self-similar curve like golden spiral in whose nature their beauty is much admired. The explanations show that source of Fibonacci numbers and how to exist Fibonacci numbers in the world we live. The mathematical theories of Fibonacci numbers and golden ratio gives the source of many new ideas in Mathematics, Chemistry, Civil engineering, Architecture, Automobile engineering, Philosophy, Botanic and biology, Electrical engineering, Computer science and engineering, Mechanical engineering, Communication systems, Mathematical education as well as theoretical physics and physics of high energy particles [1].
The Fascinating World of Real Number Sequences.pdfDivyanshu Ranjan
The study of sequences of real numbers is a fascinating and important part of mathematics. It is a central topic in analysis, a branch of mathematics that deals with continuous functions and their properties. A sequence is simply a function whose domain is the set of positive integers, and whose range is a set of real numbers. Sequences of real numbers have many interesting and useful properties, and they are used to model a wide range of mathematical and real-world phenomena.
In this book, we will delve into the properties of sequences of real numbers and explore their connections with other areas of mathematics. We will start with a gentle introduction to sequences, including the definitions and notation used in the study of sequences. We will then move on to more advanced topics, such as convergence and divergence of sequences, Cauchy sequences, and subsequences.
One of the central ideas in the study of sequences is the concept of convergence. A sequence is said to converge if its terms become arbitrarily close to a fixed real number as the index of the sequence increases. We will explore the different types of convergence and their properties, as well as the notion of limit, which is a fundamental concept in analysis.
In addition to convergence, we will also study the properties of divergent sequences, which are sequences that do not converge. We will examine the relationship between convergence and divergence and their connection with real-valued functions.
Throughout the book, we will use a friendly and engaging tone, making the material accessible to a wide range of readers, including students, mathematicians, and anyone with an interest in mathematics. Whether you are a beginner or an expert, you will find something of interest in this book.
Chapter 1: Introduction to Sequences
In this chapter, we will introduce the basic concepts and notation used in the study of sequences of real numbers. A sequence is simply a function whose domain is the set of positive integers, and whose range is a set of real numbers. We will start by defining sequences and their terms, and we will explore some basic properties of sequences, such as their limits and bounds.
We will also introduce the notation used to represent sequences, including the use of the Greek letter n to represent the index of a sequence. This is a crucial piece of notation, as it allows us to express the properties of a sequence in a concise and easily understood way.
Finally, we will look at some examples of sequences, including arithmetic and geometric sequences, and we will explore some of their properties. This will provide a foundation for the more advanced topics we will study later in the book.
Definition of Sequence
A sequence of real numbers is a function whose domain is the set of positive integers, and whose range is a set of real numbers. The elements of the sequence are referred to as terms, and they are denoted by a variable (usually "a_n") with a subscrip
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
2. Objectives:
At the end of the lesson, you should be
able to perform the following tasks:
1. Define fibonacci sequence and give its
significance.
2. Find the next terms of the given fibonacci
sequence.
3. Investigate the relationship between the
Fibonacci sequence with the golden ratio.
3. Who Was Fibonacci?
• European
Mathematician 1175-
1250
• Real name Leonardo
of Pisa.
• Author of Liber Abbaci
or Book of the Abacus
or Book of Calculation
4. What Is the Fibonacci Sequence and
Why Is It Significant?
• Generalized
sequence of first
two positive
integers and the
next number is
the sum of the
previous two, i.e.
1,1,2,3,5,8,13,21
,…
5. What Is the Fibonacci Sequence and
Why Is It Significant?
• Shows up
unexpectedly in
architecture,
science and
nature
(sunflowers &
pineapples).
6. What Is the Fibonacci Sequence and
Why Is It Significant?
• Has useful
applications with
computer
programming,
sorting of data,
generation of
random numbers,
etc.
7.
8.
9. Fibonacci Sequence
• The Fibonacci Sequence is the series of
numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
• The next number is found by adding up the two
numbers before it.
• The 2 is found by adding the two numbers
before it (1+1)
• The 3 is found by adding the two numbers
before it (1+2),
• And the 5 is (2+3), and so on!
10. Makes a Spiral
When we make squares with those
widths, we get a nice spiral:
11. The Rule
xn = xn-1 + xn-2
where:
xn is term number "n"
xn-1 is the previous term (n-1)
xn-2 is the term before that (n-2)
12. n = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
xn = 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 ...
Example: the 8th term is
the 7th term plus
the 6th term:
x8 = x7 + x6
First, the terms are numbered from 0 onwards
like this: So term number 6 is called x6 (which
equals 8).
13. n = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
xn = 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 ...
Look at the number x3 = 2. Every 3rd number is a multiple
of 2 (2, 8, 34, 144, 610, ...)
Look at the number x4 = 3. Every 4th number is a multiple
of 3 (3, 21, 144, ...)
Look at the number x5 = 5. Every 5th number is a multiple
of 5 (5, 55, 610, ...)
14. Golden Ratio
• In mathematics, two quantities are in the Golden
ratio if their ratio is the same of their sum to the
larger of the two quantities.
• “De Devina Proportione” by Luca Paciolli
15. Golden Ratio
• Is the relationship between numbers on the
Fibonacci sequence where plotting the
relationships between on scales results in a
spiral shape.
