A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers. There are a number of ways to denote a sequence, some of which are more useful for specific types of sequences. One way to specify a sequence is to list the elements. For example, the first four odd numbers form the sequence (1,3,5,7). This notation can be used for infinite sequences as well. For instance, the infinite sequence of positive odd integers can be written (1,3,5,7,...). Listing is most useful for infinite sequences with a pattern that can be easily discerned from the first few elements. Other ways to denote a sequence are discussed after the examples.
There are many important integer sequences. The prime numbers are numbers that have no divisors but 1 and themselves. Taking these in their natural order gives the sequence (2,3,5,7,11,13,17,...). The study of prime numbers has important applications for mathematics and specifically number theory. The Fibonacci numbers are the integer sequence whose elements are the sum of the previous two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is (0,1,1,2,3,5,8,13,21,34,...). Other interesting sequences include the ban numbers, whose spellings do not contain a certain letter of the alphabet. For instance, the eban numbers (do not contain 'e') form the sequence (2,4,6,30,32,34,36,40,42,...). Another sequence based on the English spelling of the letters is the one based on their number of letters (3,3,5,4,4,3,5,5,4,3,6,6,8,...). For a list of important examples of integers sequences see On-line Encyclopedia of Integer Sequences. Other important examples of sequences include ones made up of rational numbers, real numbers, and complex numbers. The sequence (.9,.99,.999,.9999,...) approaches the number one. In fact, every real number can be written as the limit of a sequence of rational numbers. For instance, the number π can be written as the limit of a sequence (3,3.1,3.14,3.141,3.1415,...). It is this fact that allows us to write any real number as the limit of a sequence of decimals. The decimal for π, however, does not have any pattern like the one for the sequence - source wikipedia Good day students in this clip we are going to be going over two examples on arithmetic sequences so lets go ahead and right down the instructions for question number one so ah for number one we are going to do the following so for the given sequence , for the given sequence a sequence is just basically a set of numbers that is separated by comas so for the given sequence we are going to do the following a you are going to um classify classify it as arithmetic or geometric as arithmetic or geometric these are the two types of sequences we have after doing that we are going to describe the growth rate and then after describing the growth rate we are going to find the next four terms after finding the next four terms we are going to find an explicit formula for the nth term ok then when we do that we will then use our explicit formula to find the 50th term alright so all these instructions we are going to apply to the sequence 5,10,15,... so this is what the sequence looks like ok it is just a list of numbers separated by commas, and then you go on forever for an infinite sequence so in this case you have an infinite sequence these dots mean that the pattern continues for eternity alright ok so lets go ahead and start with the a part part a we are to classify it as arithmetic or geometric so just remember arithmetic involves addition or subtraction geometric involves multiplication or division ok so that is the difference if you keep adding arithmetic if you keep multiplying it is geometric so adding the opposite is subtracting and multiplying the reciprocal is division there so we can just stick with addition or multiplication to separate this if we look at this sequence of numbers what is happening are we multiplying every time or are we adding? if we were multiplying let me just do some scratch work on the side 5,10,15 if we are multiplying