Arithmetic Series
An arithmetic series is the sum of the terms in an arithmetic sequence. In an arithmetic series, each term is obtained by adding a constant difference to the previous term. The sum of the first n terms of an arithmetic series can be calculated using a formula involving the first term, the last term, and the number of terms.
5 Key excerpts on "Arithmetic Series"
- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...An Arithmetic Series is the sum of the terms in the corresponding arithmetic sequence. For example, is an Arithmetic Series. To find the sum, S n, of the first n terms of an Arithmetic Series, multiply the average of the first and last terms by n. SUM OF FIRST n TERMS OF AN ARITHMETIC SEQUENCE For the arithmetic sequence 2, 5, 8, 11, 14, …, you can find the sum by using the formula, where a 1 = 2, a 36 = 107, and n = 36: Replacing a n in the formula by a 1 + (n − 1) d produces another useful formula that allows you to find the sum of the first n terms of an arithmetic sequence without first finding the n th term: SUM OF FIRST n TERMS OF AN ARITHMETIC SEQUENCE: ALTERNATIVE FORMULA To find the sum of the first 27 terms of the arithmetic sequence 2, 5, 8, 11, 14, …, use this formula with a 1 = 2, d = 3, and n = 27: EXERCISE 1 Finding the Terms of an Arithmetic Sequence An arithmetic sequence whose terms decrease in value includes 12.0 and. −2.0. If 12.0 and −2.0 are separated by three terms of this sequence, what are the values of these three terms? SOLUTION You need to determine the three numbers that make the sequence of five numbers from 12.0 to −2.0 an arithmetic sequence: To find the common difference of this sequence, use the formula a n = a 1 + (n − 1) d, where a 1 = 12, a 5 = −2.0, and n = 5: Since d = −3.5, the three terms are: EXERCISE 2 Finding the Sum of the Terms of an Arithmetic Sequence If the eighth term of an arithmetic sequence is −18 and the third term is 7, find the sum of the first 30 terms of this sequence. SOLUTION First find the common difference. Since a 3 = 7 and a 8 = −18, there are four terms between 7 and −18: • Find the common difference of the six terms of the arithmetic sequence whose first term is 7 and whose last term is −18 by using the formula a n = a 1 + (n − 1) d, where a 1 = 7, a 6 = −18, and n = 6: • Consider the original sequence, in which a 3 = 7...
- 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...
- 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...
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)
...In all, this result is obtained twelve times and gives twice the required total of the series. So the sum of the series, the amount the child will have accumulated by the end of the year, is a half of 12 (2 × 20 + 11 × 3). This comes to £438. Arithmetic progressions The example above has been an illustration of the type of series called arithmetic progressions. To generalise what was done with the particular case of the aunt and child so as to find a way of getting the sum of any arithmetic progression, we need to focus on what was fixed and what varied in the particular case. Two things were fixed - the 20 and the 3. 20 was the actual first term; it is customary to represent the first term of an arithmetic progression by the letter a. The aunt could have asked the child to save a different amount each month; a could have been given a different value. 3 was the extra, compared with the previous month, given by the aunt. It was the difference between the terms, the difference between the total accumulated in a month compared with the previous month. It is customary to represent this common difference in arithmetic progressions by the letter d. What varied from month to month was the number of ds connected to each month, but it varied from 0, for the first month, to 11 for the twelfth month. These ‘month’ numbers are connected to the total 12, the number of terms of the series. It is customary to use the letter n, in arithmetic progressions, to represent the number of terms of the arithmetic progression involved in the sum. It is also customary to use S, for sum, with a subscript n to indicate the number of terms, to represent the sum of n terms of an arithmetic progression...
