An Introduction To Sequences In Mathematics English Language Essay

An Introduction To Sequences In Mathematics English Language Essay This is an introduction to sequences. In mathematics, that is, discrete mathematics have learned about sequences, which is an ordered list of elements. The sequences is about arrangement of objects, people, tasks, grocery items, books, movies, or numbers, which has an order associated with it. Like a set, it contains members and the number of terms. This members is called elements or terms and the number of terms is also called the length of the sequences. Sequences having a natural numbers. There are all even numbers and odd numbers. This usually defined according to the formula: Sn = a, function of n = 1,2,3,a set A= {1,2,3,4} is a sequence. B = {1,1,2,2,3,3,} is though the numbers of repeating. There are specific sequences that have their own formulas and methods for finding the value of terms, such as arithmetic and geometric sequences. List of numbers, finite and infinite, that follow some rules are called sequences.P,Q,R,S is a sequences letters that differ from R,Q,P,S, as the ordering matters. Sequences can be finite or infinite. For this example is finite sequence. For example of infinite is such as the sequence of all odd positive integers (1,3,5,.). Finite sequences are sometimes known as strings or words, and infinite sequences as streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context. In this topic means sequences, there are covered about indexing, operation on sequences, sequences of integers, subsequences, increasing, decreasing, nonincreasing, nondecreasing, sigma notation, and pi notation. Besides that, in this topic also discuss about changing the index and limit in sum. Background A sequences was created by Leonardo Pisano Bigollo (1180-1250). Pisano means from Pisa and Fibonacci which means son of Bonacci. He known as by his nickname, Fibonacci. He was born in Pisa which is now part in Italy, the city with the famous Leaning Tower. He played important role in reviving ancient methematical skills, as well as making significant contributions of his own. He was known for a great interset in math. Because of the Fibonacci Series, He is most known. A series of numbers approaching nature reality. For example, 1, 1, 2, 2, 3, 5, 233, 300, 377, The sum of the 2 preceding numbers are from each succeding number. Fibonacci was a member of the Bonacci family and traveled all around the Mediterranean as a boy. He traveled with his father who held a diplomatic post. To excel in solving a wide variety of mathematical problems, His keen interest in mathematics and his exposure to other cultures allowed Fibonacci. Fibonacci is probably best known for discovering the Fibonacci sequence. Besides that, A sequences is also was created by Leonardo Fibonacci. He is the Italian mathematician. He also known as Leonardo of Pisa, documented the mathematical sequences often found in nature in 1202 in his book, Liber Abaci which means book of the abacusIn the sequences, each number is sum of two numbers, such as 1 + 1 = 2, 1 + 2 = 3, 2 + 2 = 4, and so on. That sequence can be found in the spirals on the skin of a pineapple, sunflowers, seashells, the DNA double helix and, yes, pine cones. Sequences is one such technique is a make use of Fibonacci sequences in futures. Fibonacci who was innate in 1170. He found which a settlement reoccurred in nature, as well as a settlement was subsequent from a mathematical judgment of a fibre of numbers a third series is a total of a dual prior to it. In 2000, A sequence of posters designed at the Issac Newton Institute for Mathematical Sciences which were displayed month by month in the trains of London Underground to celebrate world mathematical year 2000. The aim of the posters was to bring maths to life A sequence of posters designed at the Issac Newton Institute for Mathematical Sciences. The aim of the posters was to bring maths to life, illustrating the wide applications of modern mathematics in all branches of science includes physical, biological, technological and financial. Each poster gives relevant mathematical links and information about mathematical. Show more Result of the research A sequences is ordered list of elements that normally defined according to this formula, Sn = a function of n = 1,2,3,If S is a sequences {Sn | n = 1,2,3,},] S1 denotes the first elements, S2 denoted the second elements and so on. The indexing set of the sequences,n usually the indexing set is natural number,N or infinite subset of N. In operations on sequences, If s = { a,b,c,d,e,f } is a sequences, then -head of s = a -tail of s = {b,c,d,e,f} -tail of s = {a,b,c,d,e} -last s = f For Concatenation of sequences, If S1 = {a,b,c} and s2 = {d,e}. Hence, concatenation of s1 n s2 denoted as = {a,b,c,d,e} For this concatenation of sequences, punctuation mark , must be written between these alphabet. Increasing sequences and decreasing sequences are two important types of sequences. Their relatives are nonincreasing and nondecreasing. Sn Sn+1 is used when sequences of s is decreasing for all n for which n for which n and n+1 are in the domain of the sequences. A sequences is nonincreasing if Sn à ¢Ã¢â‚¬ °Ã‚ ¥ Sn+1 for all n for which n and n+1 are in the domain of the sequences. A sequences is nondecreasing if Snà ¢Ã¢â‚¬ °Ã‚ ¤ Sn+1 for all n for which n and n+1 are in the domain of the sequences. Example:- For increasing, Sn = 2^n 1. n= 1, 2, 3,.The first element of s are 1, 3, 5, 7,. For decreasing, Sn = 4-2^n, n = 1, 2, 3,The first few elements of s are 2, 0, -2, -4, . For nonincreasing, The sequences 100, 40, 40, 60, 60, 60, 30. For nondecreasing, the sequences of 1, 2, 3, 3, 4, 5, 5 The sequences 100, is increasing, decreasing, nonincreasing, nondecreasing since there is no value of i for which both i and i+1 are indexes. A subsequences of a sequences s is a sequences t that consists of certain elements of s retained in the original order they had in s. Example: let s = { Sn = n | n = 1,2,3,} 1,2,3,4,5,6,7,8, let t = { t=2n | n = 2,4,6,} 4, 8, 12, Hence, t is a subsequences of s. Two important operations on numerical sequences are adding and multiplying terms. Sigma notation, sum_{i=1}^{100}i. is about sum and summation. Summation is the operation of combining a sequence of numbers using addition. Hence, there are become a sum or total. Example: sum_{i=1}^ni = frac{n^2+n}2 For capital sigma notation, sum_{i=m}^n x_i = x_m + x_{m+1} + x_{m+2} +dots+ x_{n-1} + x_n. Example: sum_{k=2}^6 k^2 = 2^2+3^2+4^2+5^2+6^2 = 90 Pi is a product symbol for product of sequences of terms. This is alsoncaaled multiplication between all natural numbers. Pi notation, prod_{i=m}^n x_i = x_m cdot x_{m+1} cdot x_{m+2} cdot ,,cdots,, cdot x_{n-1} cdot x_n. Example: prod_{i=2}^6 left(1 + {1over i}right) = left(1 + {1over 2}right) cdot left(1 + {1over 3}right) cdot left(1 + {1over 4}right) cdot left(1 + {1over 5}right) cdot left(1 + {1over 6}right) = {7over 2}. Changing the index and limits in a sum. The formula to change the index and limit to the sum is, à ¢Ã‹â€ Ã¢â‚¬Ëœ_(1=0)^nà ¢-’à £Ã¢â€š ¬-ir^à £Ã¢â€š ¬-n-1 Limit of Sequence The notation of limit of a sequence is very natural. The fundamental concept of which the whole of analysis ultimately rests is that of the limit of the sequence. By considering some examples can make the position clear. Consider the sequence In this sequence, no number is zero. But we can see that the closer to zero the number of, the larger the number of n is. This state of relation can express by saying that as the number of tends to 0, the n increases, or that the sequence can converges to 0, or that they possess the limit to 0. The points crowd closer n closer to the point 0 as n increases; this means that the numbers are represented as points on a line. This situation is similar in the case of the sequence Here, too, as n increases, the numbers tends to 0; the only difference is that the numbers are sometimes less than and sometimes greater the limit 0; as we say, they oscillate about the limit. The convergence of the sequence to 0 is usually expressed by the equation or occasionally by the abbreviation . We consider the sequence where the integral index n takes all the value 1, 2, 3 à ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦. . We can see at once that as n increases, the number will approach closer and closer to the number 1 if we write, in the sense that if we mark off any interval about the point 1 all the numbers following a certain must fall in that interval. We write The sequence behaves in a similar way. This sequence also tends to a limit as n increases, to the limit 1, in symbols, . We see this most readily if we write . Here, we need to show that as n increases the number tends to 0. For all values of n greater than 2 we have and. Hence, for the remainder we have , from which at once that tend to 0 as n increases. It is also gives an estimate of the amount by which the number (for can differ maximum from the limit 1; this certainly cant exceed . The example only considered illustrates the fact to naturally expect that for large values of n the terms with the highest indices in the numerator and denominator of the fraction for predominate and that they determine the limit. Applications 1)Fibonacci number Nowadays or in era science of technology, We will find a Fibonacci number using C++ programming. The following sequences are considered: 1, 1, 2, 3, 5, 8, 13, 21, 34,.Two numbers of the sequence, a_1 and a_2 , the nth number a_n, n >=3.a_n = a_(n-1) + a_(n-2).Thus, a_3 = a_1 + a_2 = 1 + 1 = 2,a_4 = a_2 + a_3, and so on. Such a sequence is called a Fibonacci sequence. In the preceding sequence, a_2 = 1, and a_1 = 1, However any first two numbers, using this process. Nth number a_n, n >= 3 of the sequnces can be determined. The number has been determined this way is called the nth Fibonacci number. a_2 = 6 and a_1 = 3. Then, a_3 = a_2 + a_1 = 6 + 3 = 9, a_4 = a_3 + a_2 = 9 + 16 = 15. 2) Draft snake This game is most famous a long time ago. But now, a new generation still playing this game at free time. This game is closely with sequences which is about the numbers or all natural number but in this game, only positive number that have in this checker. However, it still in a sequences. Firstly, a player must play a dice to get a number so that he or she can move one place to another place to get a winner. These place to pleace is refer to the number. Each number that get from a dice will moves our position until he or she become a winner. Conclusion As we know, a sequences is about a series of numbers. A series of numbers in sequences, which is all natural number includes positive and negative integers, could be a finite sequence from some data source or an infinite sequence from a discrete dynamical system. All of the students, which is the students from the programming course learn about this topic in discrete mathematic as a minor subject in their course. Majoriti of the students said that this topic very interesting to learn and easy to score to get a highest marks in examination, test and others. Although this topic was considered very interesting to learn and easy to get a highest marks, but in this topis is also have a part that difficult to score and bored to learn. A difficult part was identified is the formula that used in this sequences. For example, one of the subtopic in a sequences is when to changing the index and limits in a sum, à ¢Ã‹â€ Ã¢â‚¬Ëœ_(1=0)^nà ¢-’à £Ã¢â€š ¬-ir^n-1à £Ã¢â€š ¬-. This formula is difficult to remembered among of the students. It is not only difficult to remembered, but a student is also difficult to remembered a way to calculate this problem where a question want a student change the index and limits in a sum. So, to solve these problem, another way must be created so that a student can solve these problem easier. May be a formula is fixed means it cannot be changed. Nowadays, a lot of ways was created by among of students to solve these problem. So another ideas must found themselves so that it easier to remembered. As a conclusion here, the subtopics in a sequences has interesting to learn and not interesting to learn. Besides that, it has easy to remembered and not easy to remembered. Here, does not all of topic are easy. This condition mest be identified so that a problem can be solved immediately and corretly among the students.

Nokia Strategic Management

Nokia’s Strategic Management Nokia Description of Company Nokia envisions a world where connecting people to what matters empowers them the most of every moment Nokia's CEO Olli-Pekka Kallasvuo Generation of Nokia NOKIA’S FIRST CENTURY: 1865-1967 †¢ The first Nokia century began with Fredrik Idestam's paper mill on the banks of the Nokianvirta river. Between 1865 and 1967, the company would become a major industrial force; but it took a merger with a cable company and a rubber firm to set the new Nokia Corporation on the path to electronics. Generation of Nokia THE MOVE TO MOBILE: 1968-1991 †¢ The newly formed Nokia Corporation was ideally positioned for a pioneering role in the early evolution of mobile communications. As European telecommunications markets were deregulated and mobile networks became global, Nokia led the way with some iconic products. Generation of Nokia MOBILE REVOLUTION: 1992-1999 †¢ As mobile phone use booms, Nokia makes the sector its core business. By the turn of the century, the company is the world leader. In 1992, Nokia decided to focus on its telecommunications business †¢ As adoption of the GSM standard grew, new CEO Jorma Ollila put Nokia at the head of the mobile telephone industry’s global boom – and made it the world leader before the end of the decade. Generation of Nokia NOKIA NOW: 2000-TODAY †¢ Nokia sells its billionth mobile phone as the third generation of mobile technology emerges. Nokia’s story continues with 3G, mobile multiplayer gaming, multimedia devices and a look to the future. Organizational Structure NAVTEQ:Manages digital map consumermobile device and marketing Nokia Siemens Network: Provides sales operational support to the units Services & Development Office. data thechannel,fixednavigation systems, Corporate Software: Develops Gives automotive network Markets: Provides supply chains, wireless and brand portfolio, Devices: Develops and manages for Internet services in 5 mobile navigation devices, messaging and games), applications, infrastructure, corporateof Internet-based mapping platforms to areas (music, maps, media,components. futureservice and worksandand andincludes communications and networks growth opportunities. activities. he sources strategic and explores government services an solutions. professional and business easily, accessible manner to consumers. deliver the services into operators and service providers. Vision of Nokia †¢ The full power of being connected †¢ Enable people to be wherever they want, whenever they want †¢ Life becomes more flexi ble and spontaneous †¢ Innovating, creating and sharing †¢ Technology becomes invisible †¢ Nokia never miss an opportunity to get the most out of life Goals of Nokia †¢ To become the leading provider of mobile solutions, because in the mobile converged internet space consumers expect seamlessly integrated solutions. To deliver these solutions requires continuous relationships with consumers and vibrant ecosystem. SWOT ANALYSIS STRENGTHS †¢Brand awareness †¢Technology leader in manufacturing mobiles †¢Market leader †¢Presence across 150 countries WEAKNESSES †¢Not good at software †¢Performance of Symbian OS is lackluster †¢Increasing dissatisfaction levels with its smartphone †¢Very weak market share in US OPPORTUNITIES †¢Huge loyal customer base †¢Huge presence in developing countries †¢Can use its infrastructure business (Nokia Siemens Network) to educe the bargaining power of mobile THREATS †¢Rapidly c hanging industry †¢Changes of missing Inflection point is high †¢Threat of entry from new business (Nokia Siemens players, Microsoft might Network) to reduce the enter smartphones market. bargaining power of mobile Google has just entered the operators market with Nexus One. Strategy Formulation Product Differentiation ? Applications are the product differentiator ? Nokia’s OVI Store ? Projection: in 2014 6. 67 billion applications would be downloaded ? Increase User Satisfaction Index ? Alliance with software developers ? Increase community strength of Maemo Strategic Objectives †¢ Irresistible solutions and vibrant ecosystems †¢ Direct and continuous consumer relationships. †¢ Best devices – Broadening their geographic reach – Broadening their device base will grow their service business †¢ Smart services – Creating relevant and personalized services – Target: 300 million people using their smart services by 2012 Strategy Formulation Cost Differentiation †¢ Nokia can set an industry enchmark for operating profits †¢ Pressure on competitors †¢ Cost leadership an invincible position against competitors †¢ Fight head-on with Apple Strategies of Nokia †¢ Competitive environment is changing †¢ Consumer needs are changing †¢ The nature of consumers’ relationships with companies is changing †¢ Irresistible solutions & vibrant ecosystem †¢ Transforming into a solutions driven company optimizing user experience. †¢ Laying the foundation for an inclusive and sustainable ecosystem †¢ Direct and continuous consumer relationships †¢ Best devices †¢ Smart services Strategies Evolution of Nokia Competitive Strategy NOKIA NOKIA Broad differentiation strategy Mass Market Low cost mass market strategy Niche Market Low cost niche market strategy Focus differentiation strategy Functional Strategy †¢ Reduce product portfolio †¢ Opportunity for targeting information users †¢ Target specific customers and specific lifestyles †¢ Connect emotionally with the target †¢ Define roadmap of Operating Systems (Symbian or Maemo) Corporate Strategy †¢ Continue divestments †¢ Concentrate resources and energy in smartphone business

Essay about Roanoke Colony - 1106 Words

The Lost Colony Jamestown is thought by most of to be the first colony in the New World but this is not the complete truth. Jamestown is considered our first successful colony; however it was not the first attempt at a colony. There were a few attempts to colonize the New World before Jamestown and one in particular that was the most mysterious is the Roanoke colony, also known as the Lost Colony. The colony got this name because the colonists that were there vanished mysteriously with no trace of what happened. Sir Humphrey Gilbert and his half-brother Sir Walter Raleigh were both veterans of earlier colony efforts. In 1578 Gilbert managed to acquire a patent from Queen Elizabeth that would let him have exclusive rights for six years†¦show more content†¦Reluctant to give up, he turned to private investors to finance another expedition. In 1585 his cousin, Sir Richard Grenville, was picked up to be the leader of a group to establish a colony on Roanoke. With him were 100 soldiers, craftsmen, and scholars to try and settle the colony. Under direction of Ralph Lane, the colony was doomed from the beginning. The settlers had arrived late in the season which made it almost impossible to plant crops and supplies were depleting and to make things worse, Lane, a captain alienated the surrounding tribes and killed their chief over a stolen cup. After Ralph Lane and his men had had enough, they left the settlement and their fort not knowing that only a week later a supply ship from England had arrived. Finding the island deserted, the leader left fifteen men behind to hold the fort and went back to England for reinforcements. In England, on January 7, 1587, a document allowing a government body to be created was signed and passed. This body was named the Governor and Assistants of the City of Roanoke in Virginia. Jon White, a skilled illustrator and map maker, was appointed Governor. On May 5, 1587 Raleigh decided to try again and boarded 117 men, seventeen women, and nine children for a more permanent settlement and sent them to the New World. The original plan was to settle at Chesapeake Bay, but on July 22, 1587 the captain, Simao Fernandes, decided toShow MoreRelatedThe Roanoke Colony On Roanoke Island911 Words   |  4 PagesThe Roanoke Colony on Roanoke Island was an attempt by Queen Elizabeth I in the late 16th-century to make a permanent English settlement in the New World. In March 1584, Queen Elizabeth granted Sir Walter Raleigh a charter for the colonization of the area of North America. This charter said that Raleigh needed to create a colony in North America, or lose his right to colonization. In April 1584, Raleigh dispatched an expedition led by Philip Amadas and Arthur Barlowe to explore the Eastern coastRead MoreEssay on The Lost Roanoke Colony741 Words   |  3 Pagesthought by most of our general population to be the first colony in the New World. This is only half true. Jamestown is considered our first successful colony, however it was not our first attempt at a colony. There were a few attempts to c olonize the New World before Jamestown and one in particular that is found to be interesting is Roanoke also known as the Lost Colony. It received this name due to the fact that the colonists that settled this colony disappeared very mysteriously. This poses the questionRead MoreThe Last Colony Of Roanoke1639 Words   |  7 PagesThe Last Colony of Roanoke Five hundred years ago, European explorers landed in North America. After trying to locate a new route to Asia across the Atlantic Ocean, they found a continent they did not know existed. Three different groups traveled to the New World, starting in 1584 (Basu, Tanya). The last group included Gov. John White, he had to return to England to submit his report to the Queen. John White tried several times to return to Virginia, but it wasn’t until a few years later he was finallyRead MoreRoanoke Island: the Lost Colony1691 Words   |  7 PagesRoanoke Island: The Lost Colony Alycia Roberts HIST113 VC On July 22, 1587, long before the Pilgrims arrived at Plymouth Rock, 117 hopeful colonists from England landed ashore onto a tiny island along the coast of what is today North Carolina. The group unpacked and founded a settlement, Roanoke Island. Then they vanished without a trace. The story of the Lost Colony has fascinated people across four centuries and remains one of the enduring mysteries of early America. There are several theoriesRead MoreThe Disappearance Of The Roanoke Island Colony1214 Words   |  5 PagesAmerica’s past is a mysterious one, riddled with unsolved questions and misleading legends. One of the most prominent enigmas that has haunted historians for centuries is the disappearance of the Roanoke Island Colony, also known to many as The Lost Colony. After leaving for three years, the governor of Roanoke Island, John White, returned to find the settlement abandon. The only remaining clue was the word â€Å"CROATOAN† carved into a fence post and the letters â€Å"CRO† etched into a nearby tree. Several diverseRead MoreEssay Lost Colony of Roanoke967 Words   |  4 Pagescenturies, the Lost Colony of Roanoke Island has been a controversial issue. Many theories exist that explain the disappearance of the colony. Some theories suggest that the colonists left the island to live with friendly neighboring Indians. Others suggest that a hurricane wiped out the colony or that a savage Indian tribe massacred them. The possibility of disease destroying them is also a debated topic. However, evidence indicates that the men and women left behind on Roanoke Island did not dieRead MoreFinding The Lost Colony Of Roanoke Essay2058 Words   |  9 PagesEnglish put forth their effort to establish in America, specifically on Roanoke island. In 1584, English colonies found east coast of North America but not permanently settled. Until 1587. Raleigh, John White and a group of 115 English settlers arrived at Roanoke Island. Although this great achievement had inflated nation s economy and promote country’s prosperity, its reign didn’t last long. John White came back to Roanoke after 3 years of disappearance. After his arrival, John had no clue whereRead MoreEssay about The Roanoke Colony3826 Words   |  16 PagesThe Roanoke colony was located on the Roanoke Island, in Dare County. This is where North Carolina is located today. In 1584, explorers Philip Amadas and Arthur Barlowe were the first Europeans to set view the island. They were sent to that particular region by Sir Walter Raleigh with the assignment of exploring the extensive sounds and estuaries in hunt of an ideal location for settlement. Barlowe wrote bright information of Roanoke Island, and when the explorers returned to England a yea r afterwardRead MoreThe Mystery Of The Lost Colony Of Roanoke2021 Words   |  9 PagesThe mystery of the Lost Colony of Roanoke is a puzzling mystery about what happened to the first English settlers in America. The question is, what actually happened to them, because even with evidence and research no one knows for absolute certain what actually did happen. The disappearance of an entire colony, who left behind a dismantled settlement and the word Croatoan etched into a tree has stumped many archaeologists. Countless theories have arisen, some more outrageous than the restRead More The Mystery of the Lost Colony of Roanoke Essay2868 Words   |  12 PagesMystery of the Lost Colony of Roanoke It was the age of discovery that first provoked intrigue and curiosity of new lands, particularly the Americas, and how the Europeans could expand to fit their society within the borders of this unknown and unexplored land. By the 1580s, more had been learned about the Americas, but any colonization until this point had not even been attempted. And so it was the English, under Queen Elizabeth Is rule, that were issued to establish a colony along the east coast