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.

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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.

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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 abacus”In 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.

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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.

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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

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-head of s = a

-tail of s = {b,c,d,e,f}

-tail of s = {a,b,c,d,e}

-last s = f

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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.

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Increasing sequences and decreasing sequences are two important types of sequences. Their relatives are nonincreasing and nondecreasing. Sn < Sn+1 is used when a sequences of s is increasing for all n for which n and n+1 are in the domain of the sequences. 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 ≥ Sn+1 for all n for which n and n+1 are in the domain of the sequences. A sequences is nondecreasing if Sn≤ 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, ….

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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.

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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,…

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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.

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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}.

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Changing the index and limits in a sum.

The formula to change the index and limit to the sum is,

∑_(1=0)^nâ-’ã€-ir^ã€-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

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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 ……. . 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 can’t 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.

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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.

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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

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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, ∑_(1=0)^nâ-’ã€-ir^n-1ã€-. 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.

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