A sequence is a function whose domain is the natural numbers. 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. The members of a sequence are called elements or terms. It runs through all the integers starting with its. 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. Sequences, summations, and recurrences wednesday, october 4, 2017 11. Each element in the series has an associated index number. Sequences, recurrence systems, series, generating functions. Summations and sequences algebra ii varsity tutors. Some useful sequences nth term first 10 terms n2 1. A sequence is a function from a subset of the z usually 0,1,2. We use the notation an to denote the image of the integer n. Sets, functions, sequences, sums, and matrices chapter 2 with questionanswer animations.
Sequences informally, a sequence is an infinite progression of objects usually numbers, consisting of a first, a second, a third, and so on. Introducing sequences in maths, we call a list of numbers in order a sequence. We use the notation a n to denote the image of the integer n. Sequences and summations vojislav kecman 19sep12 2 sequences rosen 6th ed. If and are convergent sequences, the following properties hold. Fibonacci sequence if a 0 0 and a 1 1, then nd the next ve terms. A sequence is a special type of function in which the domain is a consecutive set of integers. Arithmetic and geometricprogressions mctyapgp20091 this unit introduces sequences and series, and gives some simple examples of each. A geometric progression is a discrete analogue of the exponential function fx arx. 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. Many sequences that arise in computer science follow specific progressions. It also explores particular types of sequence known as arithmetic progressions aps and geometric progressions gps, and the corresponding series. Geometric sequences contain a pattern where a fixed amount is multiplied from one. The values of a sequence are also called terms or entries.
The final point about summations that i want to make is the notation for it. A sequence is a function whose domain is a subset of z. Project report structure fictional example revision. Sequences a sequence is a function from a subset of the set of. If the sequence is the expression is called the series associated with it. A sequence is a function from a subset of the set of integers typically the set 0,1,2. A sequence is a function from a subset of the set of. 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. The video includes of the notation that represents series and summation.
Given an arithmetic sequence, one can find the common difference by simply comparing consecutive terms in the sequence. Browse other questions tagged sequencesandseries polynomials or ask your own question. Introducing sequences infinite and finite sequences sequences and rules naming sequences types of sequence arithmetic sequences. Sequences a sequence is a function from a subset of the set of integers such as 0,1,2. Mathematics sequence, series and summations geeksforgeeks. Not surprisingly, the properties of limits of real functions translate into properties of sequences quite easily. 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.
A sequence is called a solution of a recurrence relation if its terms satisfy the recurrence relation. Nov 25, 2016 sequences and summations in discrete mathematics 1. A geometric sequence is one in which the next term is found by mutlplying the previous term by a particular constant. The variable j is referred to as the index of summation. Sequences and summations can you write an nonrecursively using n. Series and summation notation concept precalculus video. A series is simply the sum of the various terms of a sequence. Useful manipulation formulas for summation symbols are proved, and used in two illustrations. Just as with the big union and big intersection symbols, we can use this as a shorthand.
A recursion for a n is a function whose arguments are earlier. We rely on context to distinguish between a sequence and a set. 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. For example, a sequence can be defined to denote a students. 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. Instead of using the fx notation, however, a sequence is listed using the a n notation.
Geometric progression, arithmetic progression recurrence. Hyunyoung lee based on slides by andreas klappenecker 1. Sequences a sequence is an ordered list, possibly in. Thus, we look for an implicit definition which involves multiplication of the previous term. An arithmetic progression is a sequence of the form. For example, exponential growth is a growth pattern that is. This is a brief lesson on sequences and summation notation. An arithmetic progression is a discrete analogue of the linear function b. There are two standard kinds of rule for calculating a generic term a n. Summations sum of the terms from the sequence the notation. Sequences and summations cs 441 discrete mathematics for cs m.
For example, they can be used to represent solutions to certain counting problems, as we will see in chapter 8. Discrete mathematics and its applications sequences and summations lecture slides by adil aslam email me. Sequences and summations in discrete mathematics slideshare. Jan 26, 2011 intro to summation notation, with example of sum of odd numbers, and a bit about arithmetic sequences.
A sequence is a function from a subset of the set of integers usually either the set 0,1,2. Sequences and summations terms, recurrence relations, initial. It runs through all the integers starting with its lower limit m and ending with its upper limit n. The symbol for a summation is the capital greek letter sigma, which kind of looks like a backwards 3 with angles instead of curves. Finite sequences a1, a2, a3, an are called strings. Given an arithmetic sequence, one can find the common. May 09, 2011 this is a brief lesson on sequences and summation notation. There are a few examples of explicit and recursive formulas as well as partial sums and sigma notation. An arithmetic sequence is a sequence of real numbers where each term after the initial term is found by taking the previous term and adding a fixed number called. 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. Discrete structures sequences, summations, and cardinality of in nite sets 1442 example. We use the notations an or an are used to denote sequences. Sequences and summations in discrete mathematics 1. We use the notation a n to denote the image of integer n.