Arithmetic Sequences
Arithmetic sequences are a series of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference. The formula for finding the nth term of an arithmetic sequence is given by a + (n-1)d, where a is the first term and d is the common difference.
5 Key excerpts on "Arithmetic Sequences"
- Caryl Lorandini(Author)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...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. The first increase is by 2, then by 4, then 8, and then 16, so we can generalize and say that the next increase will be by 32. We can also look at any multiplication or addition that may have occurred. It appears the pattern rule of 2 n + 1 would work as well. People use patterns to predict. Patterns are analyzed to solve crimes as well. Patterns are used in designs for homes, clothes, and much more. Patterns can be very useful. 18.2 What Are Arithmetic Sequences? DEFINITIONS Sequence An ordered list of numbers. Arithmetic sequence A sequence in which the difference between any two consecutive terms is the same. So, you can find the next term in the sequence by adding the same number to the previous term. Each number is called a term in the sequence. The difference is called the c ommon difference. In this case, the common difference is +5. Sometimes you are asked to verify if a sequence is arithmetic. You would compare the terms and look for the common difference. If there is no common difference, then the sequence is not arithmetic. EXAMPLE: The common difference is −3; therefore, it is an arithmetic sequence. There is no common difference; therefore, it is not an arithmetic sequence. You can use the common difference to help you find terms in a sequence by continuing the pattern. Using the sequence 8, 5, 2, −1, −4, we can find the next few terms by knowing that the common difference is –3. EXAMPLE 18.2 1) State whether the sequence is arithmetic...
- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...A finite sequence has a definite number of terms. An infinite sequence is nonending. An infinite sequence uses three trailing periods to indicate that the pattern never ends, as in 2, 5, 8, 11, 14, …. COMMON DIFFERENCE In an arithmetic sequence, subtracting any term from the term that follows it always results in the same number. This number is called the common difference and is denoted as d. The common difference for the arithmetic sequence 2, 5, 8, 11, 14, … is 3 since A GENERALIZED ARITHMETIC SEQUENCE Consider a finite arithmetic sequence with n terms in which a 1 is the first term, a n is the n th term, and d is the common difference. The second term of this sequence is a 1 + 1 d, the third term is a 1 + 2 d, the fourth term is a 1 + 3 d, and so forth. Based on this pattern, the n th term is a 1 + (n − 1) d. n th TERM OF AN ARITHMETIC SEQUENCE For example, to find the 36th term of the arithmetic sequence 2, 5, 8, 11, 14, …, set a 1 = 2, n = 36, and d = 3: The 36th term is 107. ARITHMETIC SERIES: SUM OF TERMS The indicated sum of the terms of a sequence is called a series. An arithmetic series is the sum of the terms in the corresponding arithmetic sequence. For example, is an arithmetic series...
- 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...
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 - ePub- Valsa Koshy, Ron Casey, Paul Ernest, Valsa Koshy, Ron Casey, Paul Ernest(Authors)
- 2014(Publication Date)
- Routledge(Publisher)
...Chapter 4 Number patterns and sequences DOI: 10.4324/9780203984062-6 Objectives This chapter focuses on: 4.1 Sequences 4.2 Series 4.3 Generalised arithmetic 4.4 Functions 4.5 Identities and equations 4.6 Equations 4.7 Inequalities 4.1 Sequences Some collections of numbers exhibit a kind of pattern whereas other collections, even when arranged in different ways, seem to have no regular feature of any kind. Look at the following collection of numbers arranged in a list: 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31. What do you make of the list? It is, in fact, the list of the number of days of the months in 1999. Mathematically, it is a sequence of twelve terms; each term in the sequence has a position and a value. The term relating to April, for example, has the fourth position in the list. Its value is 30; the fourth term in the sequence is 30. What is the tenth term in the sequence? It is 31 and it relates to the month of October. Not all sequences have an obvious practical interpretation, like the months of the year. Not all sequences are finite; some are infinite - they carry on so that there is no such thing as a last term. The sequence consisting of the squares of the integers is an infinite sequence: 1, 4, 9, 16, 25, … The fourth term of this sequence, four squared, is 16. The three dots following the 25 indicate that the terms continue indefinitely. The two sequences of days and of square numbers differ in two respects. One is finite and one is infinite. The other difference concerns the connection between the position of the term in the sequence and the value of the term. 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...
