Discrete mathematics and its applications sequences and summations lecture slides by adil aslam email me. Just as with the big union and big intersection symbols, we can use this as a shorthand. We use the notation a n to denote the image of the integer n. There are a few examples of explicit and recursive formulas as well as partial sums and sigma notation. We use the notation a n to denote the image of integer n. Introducing sequences infinite and finite sequences sequences and rules naming sequences types of sequence arithmetic sequences. A geometric sequence is one in which the next term is found by mutlplying the previous term by a particular constant.
Discrete structures sequences, summations, and cardinality of in nite sets 1442 example. It also explores particular types of sequence known as arithmetic progressions aps and geometric progressions gps, and the corresponding series. We rely on context to distinguish between a sequence and a set. An arithmetic progression is a discrete analogue of the linear function b. A sequence is a function from a subset of the set of. It would not get full marks for presentation, but if youre worried about time in the test, it might give you ideas on how to complete the. Sequences and summations in discrete mathematics slideshare. Geometric sequences contain a pattern where a fixed amount is multiplied from one. Many sequences that arise in computer science follow specific progressions. For example, a sequence can be defined to denote a students gpa for each of the four years the student attended college. Sequences a sequence is an ordered list, possibly in.
A string is also denoted by a1 a2 a3 an without the commas. The video includes of the notation that represents series and summation. A sequence is a function whose domain is a subset of z. How to combinemanipulate two summations into one summation in general. A sequence is a special type of function in which the domain is a consecutive set of integers. Intro to summation notation, with example of sum of odd numbers, and a bit about arithmetic sequences. Special integer sequences arithmetic sequences are those such that consecutive differences are constant. Sets, functions, sequences, sums, and matrices chapter 2 with questionanswer animations. Sequences and summations can you write an nonrecursively using n.
For example, exponential growth is a growth pattern that is. It runs through all the integers starting with its. The members of a sequence are called elements or terms. Useful manipulation formulas for summation symbols are proved, and used in two illustrations. Each element in the series has an associated index number. Given an arithmetic sequence, one can find the common. A sequence is a function from a subset of the set of integers usually either the set 0,1,2. A sequence is a function from a subset of the z usually 0,1,2. I for such common summations, it is often useful to derive a closed form i the closed form expresses the value of the summation as a formula without summations i the closed form of above summation is. The symbol for a summation is the capital greek letter sigma, which kind of looks like a backwards 3 with angles instead of curves. Project report structure fictional example revision. Arithmetic and geometricprogressions mctyapgp20091 this unit introduces sequences and series, and gives some simple examples of each.
Not surprisingly, the properties of limits of real functions translate into properties of sequences quite easily. Summations and sequences algebra ii varsity tutors. Browse other questions tagged sequencesandseries polynomials or ask your own question. May 09, 2011 this is a brief lesson on sequences and summation notation. Finite sequences a1, a2, a3, an are called strings. We use the notations an or an are used to denote sequences. Series and summation notation concept precalculus video. Mathematics sequence, series and summations geeksforgeeks. Hyunyoung lee based on slides by andreas klappenecker 1. Jan 26, 2011 intro to summation notation, with example of sum of odd numbers, and a bit about arithmetic sequences. Instead of using the fx notation, however, a sequence is listed using the a n notation. Grieser page 5 sums of a finite arithmetic series o the sum of the first n terms of an arithmetic series is n times the mean of the first.
Fibonacci sequence if a 0 0 and a 1 1, then nd the next ve terms. Sequences and summations vojislav kecman 19sep12 2 sequences rosen 6th ed. Introducing sequences in maths, we call a list of numbers in order a sequence. This is a brief lesson on sequences and summation notation.
A recursion for a n is a function whose arguments are earlier. We use the notation an to denote the image of the integer n. If the sequence is the expression is called the series associated with it. Sequences and series lecture notes introduction although much of the mathematics weve done in this course deals with algebra and graphing, many mathematicians would say that in general mathematics deals with patterns, whether theyre visual patterns or numerical patterns. Some useful sequences nth term first 10 terms n2 1.
Sequences, summations, and recurrences wednesday, october 4, 2017 11. A geometric progression is a discrete analogue of the exponential function fx arx. Chapter 6 sequences and series in this unit, we will identify an arithmetic or geometric sequence and find the formula for its nth term determine the common difference in an arithmetic sequence determine the common ratio in a geometric sequence. Summations sum of the terms from the sequence the notation. Sequences and summations in discrete mathematics 1.
Given an arithmetic sequence, one can find the common difference by simply comparing consecutive terms in the sequence. Sequences a sequence is a function from a subset of the set of. Geometric progression, arithmetic progression recurrence. A series is simply the sum of the various terms of a sequence. Sequences and summations terms, recurrence relations, initial. The final point about summations that i want to make is the notation for it. A sequence is a function from a subset of the set of integers typically the set 0,1,2. Sequences a sequence is a function from a subset of the set integers usually 0,1,2, or 1,2,3, to a set s. It runs through all the integers starting with its lower limit m and ending with its upper limit n. Thus, we look for an implicit definition which involves multiplication of the previous term. Sequences informally, a sequence is an infinite progression of objects usually numbers, consisting of a first, a second, a third, and so on. Sequences, recurrence systems, series, generating functions. Nov 25, 2016 sequences and summations in discrete mathematics 1. Sequences a sequence is a function from a subset of the set of integers such as 0,1,2.
A sequence is a function whose domain is the natural numbers. There are two standard kinds of rule for calculating a generic term a n. If and are convergent sequences, the following properties hold. Introduction sets are one of the basic building blocks for the types of objects considered in discrete mathematics important for counting programming languages have set operations set theory is an important branch of mathematics many different systems of axioms have been used to develop set theory here we are not concerned with a formal set of axioms for. Elements can be duplicated elements are ordered a sequence is a function from a subset of z to a set s usually from the positive or nonnegative ints a n is the image of n a n is a term in the sequence a n means the entire sequence the same notation as sets. For example, they can be used to represent solutions to certain counting problems, as we will see in chapter 8. A sequence is called a solution of a recurrence relation if its terms satisfy the recurrence relation. Sequences and summations cs 441 discrete mathematics for cs m. An arithmetic progression is a sequence of the form.
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