Geometric Series
A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The sum of a geometric series can be calculated using a specific formula, which depends on the common ratio and the number of terms in the series.
7 Key excerpts on "Geometric Series"
- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...Find the sum of the first nine terms by using the formula, where, and n = 9: EXERCISE 2 Finding the Terms of a Geometric Sequence and Their Sum If the first term of a geometric sequence is 36 and the fifth term is, what is the sum of the first six terms of this sequence? SOLUTION Find the common ratio using the formula a n = a 1 r n −1, where, and n = 5: To find the sum of the first six terms of this sequence, for use the formula, where, and n = 6: You should verify that if. INFINITE Geometric Series If the common ratio, r, of a nonending geometric sequence is between 0 and 1 or between −1 and 0, the sum of its terms converges to a real number that can be determined using the following formula: SUM OF TERMS OF AN INFINITE Geometric Series, provided that | r | < 1. EXERCISE 3 Representing a Repeating Decimal as an Infinite Geometric Series Express the repeating decimal 0.131313… as the ratio of two integers. SOLUTION Because 0.131313… = 0.13 + 0.13(.01) +. 0.13(.01) 2 + 0.13(.01) 3 + …, the repeating decimal 0.131313… can be written as the sum of the terms of an infinite geometric sequence in which the common ratio is 0.01. To find the sum of the terms, use the formula, where a 1 = 0.13 and r = 0.01: Hence, the repeating decimal 0.131313… can be written in ratio form as. Lesson 15-3: Generalized Sequences KEY IDEAS Sequences can be described by formula-type expressions. 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...
- 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...
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...(2.97) and (2.98). Figure 2.5 Variation of value of n ‐term Geometric Series, S n, normalized by first term, u 0, as a function of n – for selected values of ratio, k. An alternative proof of Eq. (2.93) comes from finite induction; one should first confirm that it applies to n = 0, i.e. (2.100) Furthermore, one should prove that, if Eq. (2.93) is valid for any given n, then it will necessarily apply to n + 1, viz. (2.101) according to Eq. (2.87), one has it that (2.102) per definition – or, after splitting the summation, (2.103) Elimination of parenthesis transforms Eq. (2.103) to (2.104) where Eq. (2.87), coupled with validity of Eq. (2.93) for a given n permit transformation of Eq. (2.104) to (2.105) once u 0 is factored out, Eq. (2.105) becomes (2.106) and elimination of parenthesis produces, in turn, (2.107) Cancelation of symmetrical terms, complemented with condensation of factors alike transform Eq. (2.107) to (2.108) that coincides with Eq. (2.101) ; therefore, Eq. (2.93) will be valid for n + 1 provided that it holds for n – and one accordingly concludes, together with its validity for n = 0, that Eq. (2.93) is universally applicable to (every) n. 2.1.3 Arithmetic/Geometric Series An arithmetic/geometric sequence is the result of multiplying a geometric progression by an arithmetic progression, and looks like (2.109) in general – with k 1 serving as increment and k 2 serving as ratio, besides u 0 serving as first term; one may write Eq. (2.109) in a more condensed form as (2.110) – which is a hybrid of Eqs. (2.76) and (2.87). If both sides are multiplied by k 2, then Eq. (2.109) becomes (2.111) so ordered subtraction of Eq. (2.111) from Eq. (2.109) is in order to yield (2.112) – where parentheses were taken out for convenience; after factoring out in the left‐hand side, and condensing similar terms in the right‐hand side, Eq. (2.112) becomes (2.113) One may now factor out k 1 k 2 or (as appropriate) in Eq...
- Caryl Lorandini(Author)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...If the sequence is arithmetic, name the common difference and the next three terms. a) 2, 5, 8, 11, … b) –6, 5, 16, 27, … c), 1, 2, 4, … d) 2, 6, 18, 24, … e) 18, 11, 4, –3, … SOLUTIONS 1) a) arithmetic Common difference: +3 Next three terms: 14, 17, 20 b) arithmetic Common difference: +11 Next three terms: 38, 49, 60 c) not arithmetic d) not arithmetic e) arithmetic Common difference: –7 Next three terms: –10, –17, –24 18.3 What Are Geometric Sequences? DEFINITIONS Geometric sequence A sequence in which the quotient of any two terms is the same. So, you can find the next term by multiplying the previous term by the same number. Each number is called a term in the sequence. Common ratio The quotient. In this case, the common ratio is +5. Sometimes you are asked to verify if a sequence is geometric. You would compare the terms and look for the common ratio. If there is no common ratio, then the sequence is not geometric. EXAMPLE: The common ratio is –3; therefore, it is a geometric sequence. There is no common ratio; therefore, it is not a geometric sequence. You can use the common ratio to help you find terms in a sequence by continuing the pattern. Using the sequence 1, 2, 4, 8, 16, we can find the next few terms by knowing that the common ratio is 2. EXAMPLE 18.3 1) State whether the sequence is geometric. If the sequence is geometric, name the common ratio and the next three terms. a) 2, 6, 18, 54, … b) 25, 22, 19, 16, … c) 64, 32, 16, 8, … d) e) SOLUTIONS 1) a) geometric Common ratio: 3 Next three terms: 162, 486, 1,458 b) not geometric c) geometric Next three terms: 4, 2, 1 d) geometric e) geometric Review Exercises 1) Tell whether the sequence is arithmetic, geometric, or neither. a) 1, 3, 9, 27, 81 b) 6, 8, 10, 12, 14 c) 0, 5, 10, 15, 20 d) 1, 1, 2, 3, 5 e) 100, 10, 1,.1,.01...
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...
- Alfred Posamentier, Stephen Krulik(Authors)
- 2016(Publication Date)
- Routledge(Publisher)
...Does this distance ever reach 1? Or is there always one-half of some distance remaining? Is there a mathematical method for resolving this apparent conflict? This leads to a lesson on the infinite Geometric Series with ratio less than 1. The infinite Geometric Series with a common ratio less than one is a mathematical puzzlement to many students. Obviously, you can physically reach the doorway. However, mathematically, the sum of the sequence is never ending. The sum of the series is given by the formula, where a is the first term of the series and r is the common ratio of the Geometric Series. Here,, and. Therefore, here we have. Thus the limit of the sum of the terms in this series is 1 (the total distance from the wall to the door). Students should understand that this represents the limit of the sum of the terms in the series, and mathematically you will never reach the door. You can create a sense of amusement by continuing out the door....
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...
