Arithmetic sequence
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This document discusses arithmetic sequences and provides formulas for writing recursive and explicit expressions for terms in an arithmetic sequence. It gives an example of the arithmetic sequence 20, 24, 28, 32, 36, which has a common difference of 4. The recursive formula is defined as an = an-1 + d. The explicit or closed form formula is defined as an = a1 + (n-1)d, which allows finding any term without knowing all previous terms. It asks the reader to find the term number for the last term in the sequence -6, -3, 0, 3, ..., 147 and to determine the values of d and c in the explicit formula an = dn + c for the sequence 8, 6,
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Arithmetic sequence
This document discusses arithmetic sequences and their properties. It defines an arithmetic sequence as a sequence where the difference between consecutive terms is constant. It provides the formula for the nth term of an arithmetic sequence as an = a1 + (n-1)d, where a1 is the first term, n is the term number, and d is the common difference. It gives examples of finding specific terms and summarizing sequences. It also discusses the arithmetic mean and arithmetic sum formulas.
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This document discusses geometric sequences and provides examples. It defines a geometric sequence as a sequence where each term is found by multiplying or dividing the same value from one term to the next. An example given is 2, 4, 8, 16, 32, 64, 128, etc. The document also provides the general formula for a geometric sequence as {a, ar, ar2, ar3, ...} where a is the first term and r is the common ratio between terms. It gives practice problems for finding missing terms and the common ratio of geometric sequences.
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The document provides information about arithmetic sequences and arithmetic series. It defines an arithmetic sequence as a sequence of numbers where each term after the first is obtained by adding the same constant to the previous term. It gives examples of arithmetic sequences and explains how to find the common difference, the nth term of a sequence using the general formula, and how to solve problems involving arithmetic sequences and series. The last paragraph tells a story about how Carl Friedrich Gauss was able to quickly calculate the sum of all numbers from 1 to 100 by recognizing it as an arithmetic series.
Arithmetic Sequence
1. An arithmetic sequence is a sequence where each term is calculated by adding a fixed amount, called the common difference, to the previous term.
2. The common difference is found by subtracting the first term from the second term. This yields the constant value that is added to each term to generate the next term.
3. Formulas can be used to find a specific term in an arithmetic sequence or the sum of terms, given the first term, common difference, and number of terms.
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The document discusses arithmetic and geometric sequences and series. It defines arithmetic and geometric sequences, and provides the formulas for calculating the nth term of each type of sequence. Specifically, it states that the formula for the nth term of an arithmetic sequence is Tn = a + (n-1)d, where a is the first term and d is the common difference. The formula for the nth term of a geometric sequence is Tn = arn-1, where a is the first term and r is the common ratio. The document also provides examples of using these formulas to find terms, common differences or ratios, and to write the formulas for given sequences.
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The document discusses harmonic and Fibonacci sequences. A harmonic sequence is one whose reciprocals of terms form an arithmetic sequence. Examples of harmonic sequences are provided. The nth term of a harmonic sequence is given. It is also explained that a Fibonacci sequence is one where the first two terms are 1, and each subsequent term is the sum of the previous two. The nth term of a Fibonacci sequence cannot be determined by a single formula. Examples of Fibonacci and Fibonacci-like sequences are provided.
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The document discusses sequences and series. It defines what a sequence is, including the general term and different types of sequences such as arithmetic, finite, infinite, monotone, and piecewise sequences. It also defines arithmetic sequences specifically and provides the general term for an arithmetic sequence as an = a1 + (n - 1)d, where d is the common difference. Examples are given throughout to illustrate sequence concepts and properties.
Arithmetic Sequence
An arithmetic sequence is a sequence where each term after the first is obtained by adding a constant value, called the common difference, to the preceding term. The document provides examples of arithmetic sequences and explains how to identify them and calculate the nth term using formulas like an = a1 + (n-1)d, where a1 is the first term, d is the common difference, and n is the term number. It also shows how to use these concepts and formulas to solve problems involving finding terms and sums of arithmetic sequences.
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1. An arithmetic sequence is a sequence where each term is calculated by adding a fixed amount, called the common difference, to the previous term.
2. The common difference is found by subtracting the first term from the second term. This yields the constant value that is added to each term to generate the next term.
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The document discusses arithmetic and geometric sequences and series. It defines arithmetic and geometric sequences, and provides the formulas for calculating the nth term of each type of sequence. Specifically, it states that the formula for the nth term of an arithmetic sequence is Tn = a + (n-1)d, where a is the first term and d is the common difference. The formula for the nth term of a geometric sequence is Tn = arn-1, where a is the first term and r is the common ratio. The document also provides examples of using these formulas to find terms, common differences or ratios, and to write the formulas for given sequences.
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The document discusses sequences and series. It defines what a sequence is, including the general term and different types of sequences such as arithmetic, finite, infinite, monotone, and piecewise sequences. It also defines arithmetic sequences specifically and provides the general term for an arithmetic sequence as an = a1 + (n - 1)d, where d is the common difference. Examples are given throughout to illustrate sequence concepts and properties.
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An arithmetic sequence is a sequence where each term after the first is obtained by adding a constant value, called the common difference, to the preceding term. The document provides examples of arithmetic sequences and explains how to identify them and calculate the nth term using formulas like an = a1 + (n-1)d, where a1 is the first term, d is the common difference, and n is the term number. It also shows how to use these concepts and formulas to solve problems involving finding terms and sums of arithmetic sequences.
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This module introduces sequences and their different types. It discusses finding patterns in sequences to determine the next term. Specific examples are provided to demonstrate writing the first few terms of a sequence given its general formula. The key concepts covered are defining sequences, finite vs infinite sequences, terms of a sequence, increasing vs decreasing sequences, and using the general formula to find specific terms. Students are expected to be able to list terms of sequences, derive the formula, generate terms recursively, and describe arithmetic sequences in different ways.
Arithmetic sequences and series
This document introduces arithmetic sequences and series. It defines an arithmetic sequence as a sequence where the difference between consecutive terms is constant. The nth term of an arithmetic sequence can be written as a formula of a, the first term, and d, the common difference. An arithmetic series is the sum of terms in an arithmetic sequence, which can be calculated using the sigma notation and the formula for the sum of an arithmetic sequence. Examples are provided to illustrate finding terms and sums of arithmetic sequences and series.
Arithmetic progression
The document discusses arithmetic progressions (AP), which are sequences where the difference between successive terms is constant. It defines an AP using the notation an = a + (n-1)d, where a is the first term, d is the common difference, and n indexes the terms. It provides examples and explains how to find individual terms, determine if a sequence is an AP, and calculate the sum of terms in an AP using the formula Sn = 1/2n(2a + (n-1)d). Various problems are worked through to demonstrate applying the concepts and formulas for APs.
Arithmetic And Geometric Progressions
This document introduces sequences and series, focusing on arithmetic and geometric progressions. It provides:
1) Definitions of sequences, series, and the differences between them. Arithmetic progressions have a common difference between terms, while geometric progressions multiply the previous term by a common ratio.
2) Formulas for calculating terms and sums of arithmetic and geometric progressions. The sum of an arithmetic progression can be expressed in terms of the first term, last term, number of terms, and common difference.
3) Examples of using the formulas to find specific terms and sums of progressions. Practice with these types of problems is important to master the techniques.
Ap gp
This document introduces arithmetic and geometric progressions. It defines a sequence as a set of numbers written in a particular order. A series is the sum of the terms in a sequence. An arithmetic progression is a sequence where each new term is obtained by adding a constant difference to the preceding term. The sum of an arithmetic progression can be found using the formula: the sum of the first n terms is equal to one-half n times the quantity of two times the first term plus (n - 1) times the common difference.
M3 l1 sequences & series
The document discusses sequences and series. It defines explicit and recursive sequences, and provides examples of finding terms of each type of sequence. It also defines finite and infinite series, and explains how to determine whether a series converges or diverges based on whether the sum of the terms approaches a finite number. Examples are provided of identifying whether series are finite or infinite, and determining if they converge or diverge. Summation notation is introduced as a way to write series mathematically. Overall, the document provides an introduction to key concepts about sequences and series.
sequence and series
The document defines sequences and series. It explains that a sequence is an ordered list of numbers with a specific pattern, while a series is the sum of the terms in a sequence. The document provides examples of arithmetic and geometric sequences, and explains how to determine the nth term of each using formulas involving the first term, common difference or ratio, and position of the term. It also discusses finite vs infinite sequences and gives examples of sequences in real world contexts like running training and loan interest.
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Notes 12.1 Arithmetic Sequences and Series.pdf
This document provides information about arithmetic sequences and series. It defines key terms like sequence, term, and common difference. It also presents methods for determining subsequent terms in a sequence, the nth term of a sequence using the explicit formula, finding arithmetic means between terms, and evaluating finite arithmetic series using the summation formula. Application examples are provided to demonstrate how to use these concepts to solve real-world problems involving sequences and series.
I am kind of confused about quantifiers. I am not sure how to transl.pdf
I am kind of confused about quantifiers. I am not sure how to translate them into english
correctly so a quick Guide to how or the answer will work. Thanks!
***** (A = universal) , (E = existential), (V = or), (^ = and) I do not know how to input the real
symbol)
1.) AxAyAz([x+y]+z = x+([y+z])
2.) Here x, y are students C(x) is having a computer and F(x,y) indicates that x and y are friends.
Ax(C(x) vEy[C(y)^F(x,y)]
3.) Suppose that x and y are real numbers
a. AxAy([x>0 ^ y>0] ------> [x+y>0])
b. AxAy([x<0 ^ 4<0] [x(y)>0])
Solution
There are several logical symbols in the alphabet, which vary by author but usually
include: The quantifier symbols and The logical connectives: for conjunction, for disjunction, for
implication, for biconditional, for negation. Occasionally other logical connective symbols are
included. Some authors use , or Cpq, instead of , and , or Epq, instead of , especially in contexts
where is used for other purposes. Moreover, the horseshoe may replace ; the triple-bar may
replace , and a tilde (~), Np, or Fpq, may replace ; ||, or Apq may replace ; and &, or Kpq, may
replace , especially if these symbols are not available for technical reasons. Parentheses,
brackets, and other punctuation symbols. The choice of such symbols varies depending on
context. An infinite set of variables, often denoted by lowercase letters at the end of the alphabet
x, y, z, … . Subscripts are often used to distinguish variables: x0, x1, x2, … . An equality symbol
(sometimes, identity symbol) =; see the section on equality below. It should be noted that not all
of these symbols are required - only one of the quantifiers, negation and conjunction, variables,
brackets and equality suffice. There are numerous minor variations that may define additional
logical symbols: Sometimes the truth constants T, Vpq, or , for \"true\" and F, Opq, or , for
\"false\" are included. Without any such logical operators of valence 0, these two constants can
only be expressed using quantifiers. Sometimes additional logical connectives are included, such
as the Sheffer stroke, Dpq (NAND), and exclusive or, Jpq. [edit]Non-logical symbols The non-
logical symbols represent predicates (relations), functions and constants on the domain of
discourse. It used to be standard practice to use a fixed, infinite set of non-logical symbols for all
purposes. A more recent practice is to use different non-logical symbols according to the
application one has in mind. Therefore it has become necessary to name the set of all non-logical
symbols used in a particular application. This choice is made via a signature.[2] The traditional
approach is to have only one, infinite, set of non-logical symbols (one signature) for all
applications. Consequently, under the traditional approach there is only one language of first-
order logic.[3] This approach is still common, especially in philosophically oriented books. For
every integer n = 0 there is a collection of n-ary, or n-place, predicate .
Stochastic Processes Homework Help
I am Simon M. I am a Stochastic Processes Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Stochastic Processes, from Texas, USA. I have been helping students with their homework for the past 7 years. I solve assignments related to Stochastic Processes. Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com. You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
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The document discusses the key differences between the English and mathematics languages, including how words, symbols, and concepts have different meanings or representations. It explains how mathematics has developed a precise symbolic language to concisely express relationships, operations, and concepts in a way that is internationally understood regardless of spoken language. Precise definitions and notations are provided for important mathematical concepts like sets, relations, functions, and equations.
Mc ty-surds-2009-1
This document discusses simplifying expressions involving surds, which are irrational numbers that cannot be written as fractions such as √2. It provides techniques for manipulating surds, including:
1) Simplifying expressions using properties of square roots and rationalizing denominators with surds to remove irrational terms;
2) Using the difference of squares formula to expand expressions like (a + b)(a - b);
3) Rationalizing fractions with surds in the denominator to obtain equivalent fractions with whole number denominators.
arithmetic sequences explicit.ppt
The document provides information about arithmetic sequences including defining the first term (a), common difference (d), and the explicit formula to find any term (an) in an arithmetic sequence. It gives examples of finding specific terms and writing recursive and explicit rules for arithmetic sequences.
Stochastic Processes Homework Help
I am Kennedy, G. I am a Stochastic Processes Assignment Expert at excelhomeworkhelp.com. I hold a Ph.D. in Stochastic Processes, from Indiana, USA. I have been helping students with their homework for the past 7 years. I solve assignments related to Stochastic Processes. Visit excelhomeworkhelp.com or email info@excelhomeworkhelp.com. You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
Linear equation class 8 by siddhartha kumar panda
This document provides an overview of linear equations in one variable. It defines a linear equation as one that can be written in the form ax = b, where a and b are real numbers and a ≠ 0. Parts of a linear equation like the variable, coefficient, constants, and left and right hand sides are explained. It also discusses how to solve linear equations by finding the value of the variable that makes the left and right sides equal. Several examples are provided of how to translate word problems into linear equations and solve them. Real-world applications of linear equations discussed include comparing rates of pay from different jobs and calculating cab fares.
9.4 part 1 and 2 combined worked
This document provides information about number patterns, sequences, and series. It defines recursive and explicit formulas for sequences and discusses arithmetic and geometric sequences. For arithmetic sequences, the common difference is constant between terms. For geometric sequences, the common ratio between consecutive terms is constant. Formulas are provided to find later terms in sequences, whether defined recursively or explicitly. Examples are given to practice identifying sequence types and using the formulas to find missing or future terms.
9.2 Arithmetic Sequences
* Find the common difference for an arithmetic sequence.
* Write terms of an arithmetic sequence.
* Use a recursive formula for an arithmetic sequence.
* Use an explicit formula for an arithmetic sequence.
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Grade 10 Math Module 1 searching for patterns, sequence and series
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Arithmetic sequence
- 1. ARITHMETIC SEQUENCE BY: LEAH V. MELENDRES By PresenterMedia.com
- 2. This lesson will work with arithmetic sequences, their recursive and explicit formulas and finding terms in a sequence. In this lesson, it is assumed that you know what a sequence is.
- 3. Let’s look at the arithmetic sequence 20, 24, 28, 32, 36, . . . This arithmetic sequence has a common difference of 4, meaning that we add 4 to a term in order to get the next term in the sequence.
- 4. The recursive formula for an arithmetic sequence is written in the form an = an-1 + d in our example, since the common difference (d) is 4, we would write: an = an-1 + 4
- 5. So once you know the common difference in an arithmetic sequence you can write the recursive form for that sequence. However, the recursive formula can become difficult to work with if we want to find the 50th term. Using the recursive formula, we would have to know the first 49 terms in order to find the 50th. This sounds like a lot of work. There must be an easier way. And there is!
- 6. Rather than write a recursive formula, we can write an explicit formula. Explicit formula is also sometimes called the closed form. To write the explicit or closed form of an arithmetic sequence, we use an = a1 + (n-1)d an – is the nth term of the sequence. When writing the general expression for an arithmetic sequence, you will not actually find a value for this. (It will be part of your much in the same way x’s and y’s are part of algebraic equations). a1 – is the first term in the sequence. n – is treated like the variable in a sequence. d – is the common difference for the arithmetic sequence. You will either be given this value or be given enough information to it. You must substitute a value for d into the formula
- 8. Given an arithmetic sequence:-6, -3, 0, 3, ..., 147.What is the term number for the last number in the sequence? [a? = 147] Given: 8, 6, 4, 2, ...For the formula that represents the general term, an = dn + c, what are the correct values for d and c ?
- 9. THANK YOU!!!