Pre-Cal 40S Slides May 29, 2007
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The document defines sequences and provides information about arithmetic and geometric sequences. It explains that: - An arithmetic sequence follows a pattern where each term is generated by adding a common difference to the previous term. The common difference is the slope of the linear equation that generates the terms. - The nth term in an arithmetic sequence can be calculated as: tn = a + (n - 1)d, where a is the first term, n is the position of the term, and d is the common difference. - A geometric sequence follows a pattern where each term is generated by multiplying the previous term by a common ratio. The common ratio is the base of the exponential equation that generates the terms.
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The document discusses sequences and provides examples of arithmetic and geometric sequences. It defines key terms like sequence, arithmetic sequence, common difference, geometric sequence, and common ratio. Formulas are given for finding the nth term in an arithmetic sequence and a geometric sequence. Examples are shown for calculating the common difference and finding a specific term in an arithmetic sequence.
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The document defines sequences and series in precalculus. A sequence is a function with positive integers as its domain. A series represents the sum of the terms of a sequence. The document provides examples of arithmetic and geometric sequences, and defines their associated series. It also discusses infinite geometric series and harmonic sequences. Examples are given to identify sequences and series, determine sequence terms, and identify types of sequences.
Pre-Cal 40S June 4, 2009
The document discusses arithmetic and geometric sequences and series. It provides the formulas to find the nth term of an arithmetic sequence and geometric sequence. Examples are given to find specific terms. It also discusses how to calculate the sum of terms in an arithmetic series and geometric series using the summation notation. Formulas are given for the sum to the nth term of an arithmetic series and geometric series.
13 1 arithmetic and geometric sequences
This document discusses arithmetic and geometric sequences. It defines arithmetic sequences as having a constant difference between consecutive terms, called the common difference. Geometric sequences have a constant ratio between consecutive terms, called the common ratio. Formulas are provided for finding the nth term of an arithmetic sequence and a geometric sequence based on the initial term and common difference or ratio. Examples are given of identifying the type of sequence and calculating terms. The document also discusses notation, formal definitions, and graphing sequences.
Arithmetic seqence
The document defines an arithmetic sequence as a sequence where a constant difference can be added between each term to get the next term. It provides examples of finding the first term and common difference of arithmetic sequences. It explains that the common difference is found by subtracting any term from the one that follows it. Formulas are provided for finding specific terms in an arithmetic sequence given the first term, common difference, or other information. Examples problems are worked through applying these concepts.
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.
Patterns
The document discusses different types of numerical patterns, including arithmetic sequences and geometric sequences. An arithmetic sequence is a list of numbers where each subsequent term is calculated by adding a fixed value called the common difference to the previous term. A geometric sequence is a list where each term is calculated by multiplying the previous term by a fixed value called the common ratio. Formulas are provided to calculate the nth term in an arithmetic or geometric sequence given the first term, common difference or ratio, and n. Examples are worked through applying the formulas to find specific terms in sample sequences.
Sequences and series
The document discusses arithmetic and geometric sequences and series. It provides examples of different sequences and explains how to determine the general formulas for finding any term in the sequence. Specifically, it explains that arithmetic sequences have a common difference and use the formula un = u1 + (n-1)d, while geometric sequences have a common ratio and use the formula un = u1rn-1. It also discusses how series are used to sum the terms of sequences and provides examples of applications for arithmetic and geometric sequences/series such as solving math problems, determining total costs or earnings, and calculating compound interest or population growth.
Arithmetic Sequences and Series-Boger.ppt
The document discusses arithmetic sequences and series. Some key points include:
- An arithmetic sequence is a sequence whose consecutive terms have a common difference. The common difference is the number added to get the next term.
- An arithmetic series is the sum of terms in an arithmetic sequence.
- The formula to find any term in an arithmetic sequence is an = a1 + (n - 1)d, where a1 is the first term, n is the term number, and d is the common difference.
- The formula to find the sum of the first n terms of an arithmetic sequence is Sn = (n/2)(a1 + an), where a1 is the first term, an is the last
Arithmetic Sequences and Series-Boger.ppt
The document discusses arithmetic sequences and series. Some key points include:
- An arithmetic sequence is a sequence whose consecutive terms have a common difference. The common difference is the number added to get the next term.
- An arithmetic series is the sum of terms in an arithmetic sequence.
- Formulas are provided for finding the nth term of an arithmetic sequence and the sum of the first n terms of an arithmetic sequence.
- Examples are given of finding terms, common differences, and sums of arithmetic sequences and series.
Arithmetic Sequences and Series-Boger.ppt
The document discusses arithmetic sequences and series. It provides examples of arithmetic sequences and how to determine the common difference and write terms. It also discusses how to find individual terms, sums of terms, and arithmetic means within sequences. Formulas are provided for the nth term of a sequence, the sum of terms, and evaluating sequences.
Applied 20S January 7, 2009
The document defines arithmetic sequences as lists of numbers where each term is generated by adding a common difference to the previous term. It provides examples of arithmetic sequences and explains how to find the nth term of a sequence using the first term and common difference. Several word problems involving arithmetic sequences are also presented along with their solutions.
Sequences, Series, and the Binomial Theorem
An infinite sequence is a function whose domain is the set of natural numbers, while a finite sequence has a domain of natural numbers up to some limit. A sequence can be described by its general term, which gives a rule for calculating each term based on its position in the sequence. The sum of the terms of a sequence is called a series, which is finite if it includes a finite number of terms and infinite if it includes all terms.
Sequences and series
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.
Sequences
This document discusses different types of sequences:
1. Arithmetic sequences are sequences where the difference between consecutive terms is constant. The nth term can be calculated using a formula involving the first term and the common difference.
2. Geometric sequences are sequences where the ratio between consecutive terms is constant. The nth term can be calculated using a formula involving the first term and the common ratio.
3. Examples are provided to demonstrate how to calculate the nth term and sum of terms for both arithmetic and geometric sequences. Formulas are given for finding individual terms and summing finite and infinite sequences.
Gg
The document defines key concepts related to sequences and series:
- An infinite sequence is a function with positive integers as its domain. Geometric sequences have a common ratio between consecutive terms.
- The nth term of a geometric sequence is given by an = a1rn-1, where a1 is the first term and r is the common ratio.
- A geometric series is the sum of terms of an infinite geometric sequence. If the common ratio r satisfies |r|<1, the sum of the infinite series can be calculated as a1/(1-r).
- Examples demonstrate calculating individual terms, sums of finite sequences, and sums of infinite series using the given formulas.
Task4 present
The document discusses different types of sequences including arithmetic, geometric, special integer sequences like triangular numbers, square numbers, Fibonacci numbers and cube numbers. It provides examples and definitions of each type. It also covers the principles of mathematical induction and how it is used to prove statements for all positive integers, involving two steps: basis step to verify the statement is true for n=1, and inductive step to show the statement is true for n+1 assuming it is true for n. An example is provided to demonstrate a proof using mathematical induction.
Sequences finding a rule
The document discusses sequences, which are sets of terms in a definite order obtained by some rule. It provides examples of finite and infinite sequences and discusses notation for representing sequences. It also covers finding the formula for the terms of a sequence using recurrence relations, which define the initial terms and the relationship between successive terms. The document gives examples of identifying patterns in sequences and finding subsequent terms.
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.
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- 1. Sequences Photo source chromaspiral
- 2. Find the next three terms in each sequence of numbers ... 4, 7, 10, 13, , , 3, 6, 12, 24, , , 32, 16, 8, 4, , , 1, 1, 2, 3, 5, 8,13, , ,
- 3. 4, 7, 10, 13, , ,
- 4. Sequence: An ordered list of numbers that follow a certain pattern (or rule). Arithmetic Sequence:(i) Recursive Definition: An ordered list of numbers generated by continuously adding a value (the common difference) to a given first term. (ii) Implicit Definition: An ordered list of numbers where each number in the list is generated by a linear equation. Common Difference (d):(i) The number that is repeatedly added to successive terms in an arithmetic sequence. (ii) From the implicit definition, d is the slope of the linear equation.
- 5. To Find The Common Difference d = tn - t(n - 1) d is the common difference tn is an arbitrary term in the sequence t(n - 1) is the term immediately before tn in the sequence To Find the nth Term In an Arithmetic Sequence tn = a + (n - 1)d tn is the nth term a is the first term n is the quot;rankquot; of the nth term in the sequence d is the common difference Example: Find the 51st term (t51) of the sequence 11, 5, -1, -7, ... Solution: a = 11 t51 = 11 + (51 - 1)(-6) d = 5 - 11 t51 = 11 + (50)(-6) = -6 t51 = 11 - 300 n = 51 t51 = -289
- 6. 3, 6, 12, 24, , , 32, 16, 8, 4, , ,
- 7. Geometic Sequence: (i) Recursive Definition: An ordered list of numbers generated by continuously multiplying a value (the common ratio) with a given first term. (ii) Implicit Definition: An ordered list of numbers where each number in the list is generated by an exponential equation. Common Ratio (r): (i) The number that is repeatedly multiplied to successive terms in a geometic sequence. (ii) From the implicit definition, r is the base of the exponential function.
- 8. To Find The Common Ratio r is the common ratio tn is an arbitrary term in the sequence t(n - 1) is the term immediately before tn in the sequence To Find the nth Term In a Geometic Sequence tn is the nth term a is the first term n is the quot;rankquot; of the nth term in the sequence r is the common ratio