Generating Terms of a Sequence
Generating terms of a sequence involves finding the individual elements of a sequence based on a specific rule or pattern. This can be done by applying the given rule to each term in the sequence to generate the subsequent terms. The process of generating terms of a sequence is fundamental in understanding patterns and relationships in mathematics.
6 Key excerpts on "Generating Terms of a Sequence"
- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...The sum of any number of consecutive terms of a generalized sequence can be indicated by using the symbol Σ, the capital Greek letter sigma. VIEWING A SEQUENCE AS A FUNCTION A sequence is a function whose domain is a set of consecutive whole numbers that represent the position numbers of the terms in the sequence and whose range is the corresponding terms of the sequence. Instead of using standard function notation such as a (1) to indicate the first term of the sequence, a (2) to indicate the second term, and so forth, the terms or function values of a sequence are represented by the subscripted variables a 1, a 2, a 3, …, a n, …, where the position of a term is indicated by its subscript. The notation { a n } refers to the entire sequence in which a n is a general term. For the sequence 10, 15, 20, 25, and 30, the domain is the set of position numbers, {1, 2, 3, 4, 5}, and the range is the corresponding set of terms, {10, 15, 20, 25, 30}. If a represents this sequence function, then a 1 = 10, a 2 = 15, a 3 = 20, a 4 = 25, and a 5 = 30. EXPLICIT VERSUS RECURSIVE FORMULAS A sequence may be defined explicitly or recursively. • A sequence is defined explicitly when a formula is given that tells how to obtain all of the terms of the sequence without knowing the identity of any specific term. The formula a n = 2 n + 1 defines a sequence explicitly since it tells how to obtain each term without knowing any other term: • A sequence is defined recursively when all of its terms can be obtained by a statement that relates a general term of the sequence to one or more terms of the sequence that preceded it. Suppose that a 1 represents the initial balance in year 1 of a savings account in which interest is compounded yearly at the rate of 5%...
eBook - ePub- Christine Hopkins, Ann Pope, Sandy Pepperell(Authors)
- 2013(Publication Date)
- David Fulton Publishers(Publisher)
...This approach looking at the structure of the problem is often helpful in finding a formula for the n th term. Another way of defining the sequence is to show how each term is derived from the one before. In this sequence the next shape is made by adding one extra circle at the end of each arm, that is adding four circles. The first term in the sequence is 5. The number sequence goes 5, 9, 13, 17, 21… Taking the last term and adding four will give the next term in the sequence. This is called a term-to-term or an inductive definition. DIFFERENCE METHOD If you have a sequence which goes up by a constant amount each time it is possible to work out the formula for the n th term. The sequence goes up in 3s just like the multiples of 3. 7 10 13 16 19 … sequence 3 6 9 12 15… multiples of 3 Comparing the sequence with the multiples of 3, each value in the sequence is 4 more than the corresponding multiple of 3. As the formula for the multiples of 3 is 3 n, the formula for the sequence is 3 n + 4. Similarly for the sequence 3, 8, 13, 18, 23 … The sequence goes up in 5s just like the multiples of 5. 3 8 13 18 23 … sequence 5 10 15 20 25… multiples of 5 Each value in the sequence is 2 less than the corresponding value in the multiples of 5. The formula for the n th term of the multiples of 5 is 5 n. So formula for the sequence is 5 n − 2. Sequences with a constant difference are called linear sequences. Consider the sequence of square numbers The differences are not constant, they go up by 2 each time. A sequence...
eBook - ePubGMAT Advanced Quant
250+ Practice Problems & Online Resources
- (Author)
- 2020(Publication Date)
- Manhattan Prep(Publisher)
...Sequence Problems Any question that involves the definition of a sequence (usually involving subscripted variables, such as A n and S n) is very likely to involve patterns. These patterns can range from relatively straightforward linear patterns to much more complicated ones. When you are given a sequence definition, list a few terms of the sequence, starting with any particular terms you are given, and look for a pattern. Do not be intimidated by a recursive definition for a sequence, in which each term is defined using earlier terms. (By contrast, a direct definition defines each term using the position or index of the term.) To illustrate the difference, here are two ways to define the series of positive odd integers {1, 3, 5, 7, 9, etc.}: Recursive Definition Direct Definition A n = A n − 1 + 2 where n > 1 and A 1 = 1 Translation: “This term = the previous term + 2, and the first term is 1.” A n = 2 n − 1, where n ≥ 1 Translation: “This term = the index number × 2, minus 1. Thus, the first term is (2)(1) − 1 = 1.” Try-It #5-2 The sequence X n is defined as follows: X n = 2 X n − 1 − 1 whenever n is an integer greater than 1. If X 1 = 3, what is the value of X 20 − X 19 ? The pattern underlying this sequence is not obvious, so begin computing a few of the terms in the set: n X n 1 3 2 2(3) − 1 = 5 3 2(5) − 1 = 9 4 2(9) − 1 = 17 5 2(17) − 1 = 33 6 2(33) − 1 = 65 7 2(65) − 1 = 129 You might notice that there appears to be a repeating pattern among the units digits of the elements of X n (3, 5, 9, 7, 3, 5, 9. . .). However, this does not help to answer the question, which asks about the difference between two consecutive elements later in the set...
- Caryl Lorandini(Author)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...18 SEQUENCES WHAT YOU WILL LEARN • How to find the terms of arithmetic sequences • How to find the terms of geometric sequences SECTIONS IN THIS CHAPTER • What Are Patterns? • What Are Arithmetic Sequences? • What Are Geometric Sequences? 18.1 What Are Patterns? DEFINITIONS Numeric pattern An arrangement of numbers that repeat or follow a specified rule. Pattern A design (geometric) or sequence (numeric or algebraic) that is predictable because some aspect of it repeats. It is only natural for us to look for patterns in things. Nature is full of patterns. The triangle numbers are numbers generated from a pattern of dots that form a triangle. The pattern looks like this: The numbers start 1, 3, 6, 10, 15, 21, 28, 36, … The square numbers are numbers generated from a pattern of dots that form squares. You can also find them by squaring. The numbers start 1, 4, 9, 16, 25, 36, 49, … There are many other patterns. The next number is found by cubing. The next number is found by adding the two preceding numbers together. Some patterns are sequences, or sets of things in order. The things in the sequence are called members or elements. If the sequence goes on forever, it is called an infinite sequence; otherwise, it is called a finite sequence. Sequences also use the same notation as sets use: list each element, separated by a comma, and then put brackets around the whole thing. EXAMPLE: Even whole numbers: A sequence usually has a rule, which is a way of finding the value of each term. For example, we found the even whole numbers by starting with zero and adding two to each preceding term. Let’s look at some sequences to try to find the pattern and the rule. EXAMPLE: We know the numbers are increasing...
eBook - ePub- Valsa Koshy, Ron Casey, Paul Ernest, Valsa Koshy, Ron Casey, Paul Ernest(Authors)
- 2014(Publication Date)
- Routledge(Publisher)
...If you have ‘seen’ the pattern in the sequence of squares of the integers, you should be able to give the seventh term in the sequence. You take the 7 and square it to obtain 49. There is a rule which connects the position of the term and its value. There is no rule which connects the number of the month and the number of the days in the month. For October, the tenth month, there is not a rule which enables you to obtain 31 so that applying the same rule in the same way to 6, for June, you can obtain 30. Of course, there is a constancy in the list of days in the months of the year; only the number for the second month varies and most people know of the constancy. But the constancy is not a rule in the way that a rule applies to the sequence of squares. Sequences which can be generated by a rule are of more interest to mathematicians than sequences of numbers which are not generated by a rule. The latter type of sequence is really just a list; it could be a list of data items obtained in an experiment or even a list of random numbers for use on a Lottery ticket. Working out sequences There are two skills associated with sequences governed by a rule: 1 The skill which enables someone, given the rule, to find terms of the sequence specified by the position number of the terms. For example, 4, 8, 12,16,…; here the number that fits into blank space is 20 which is the fifth in the sequence. This is obtained by multiplying 5 × 4 = 20, the 20th term will be 20 × 4 and following this rule the nth term will be n × 4 or 4n. 2 The skill which enables someone, given at least three consecutive terms of the sequence, to discover the rule for the sequence so that further terms can be generated. Perhaps the simplest of rules is ‘add 1 to each term to obtain the next term’. Following this rule enables someone to count on from any starting number. Beginning with 1, using the rule generates 2, 3, 4, 5 and so on...
- J. Rosebush, Stu Schwartz(Authors)
- 2016(Publication Date)
- Research & Education Association(Publisher)
...PART V SEQUENCES AND SERIES Chapter 16 Sequences and Series I. SEQUENCES —a sequence is a list of numbers separated by commas a 1, a 2, a 3,..., a k,..., that may or may not have a pattern. A. Arithmetic and geometric sequences 1. The formula for the n th term of an arithmetic sequence (one that is formed by adding the same constant repeatedly to an initial value) is a n = a 1 + (n – 1) d where a 1 is the first term of the sequence, n is the number of terms in the sequence, and d is the common difference. The formula for the n th term of a geometric sequence (one that is formed by multiplying the same constant repeatedly to an initial value) a n = a 1 r (n –1) where a 1 is the first term, r is the common ratio, and n is the number of terms in the sequence. 2. Convergent sequences—a sequence converges if it approaches a number. A sequence can be thought of as a function whose domain is the set of positive integers. As such, the concept of limit of a sequence is the same as the concept of limit of a function. 3. Divergent Sequences—a sequence is divergent if it does not approach a particular number; that is, it approaches ±∞. II. SERIES —a series is the sum of the terms of a sequence. A series converges if the sequence of its partial sums converges. For the sequence of partial sums is given by where S 1 = a 1, S 2 = a 1 + a 2, S 3 = a 1 + a 2 + a 3,..., S k = a 1 + a 2 + a 3 + … + a k. With most series, it is possible only to figure out whether it converges (or diverges) but not to figure out the actual sum. In general, the series for which it is possible to find the sum, if it exists, are geometric series and telescoping series. A. Types of infinite series 1. Geometric series —this series is of the form This series converges (that is, its sum exists) if and only if | r | < 1 (that is, –1 < r < 1). If it converges, its sum is given by. 2. p - series,, converges when p > 1 and diverges when 0 < p ≤ 1. 3. Alternating series are series with terms whose signs alternate...
