Arithmetic and geometric_sequences
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The document discusses arithmetic sequences and provides examples of determining whether a sequence is arithmetic, writing recursive and explicit formulas for arithmetic sequences, and finding terms of arithmetic sequences given a formula. It includes examples of identifying the common difference of an arithmetic sequence, writing recursive and explicit formulas, and using formulas to find specific terms. Formulas shown include the recursive formula A(n) = A(n-1) + d and explicit formula A(n) = A(1) + (n-1)d for an arithmetic sequence.
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The document discusses arithmetic and geometric sequences. It provides examples and explanations of key concepts such as common difference, common ratio, formulas for finding specific terms, and calculating series sums. Formulas are derived for finding the nth term in an arithmetic sequence as an = a1 + d(n-1) and the nth term in a geometric sequence as an = a1rn-1. Examples are worked through to demonstrate how to use the formulas and calculate sequence terms and series sums.
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For class 10.
In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant
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- Sequences show a listing of numbers in a pattern, while series express the sum of the terms in a sequence.
- Arithmetic sequences have a common difference between consecutive terms, and their series is calculated using the formula Sn = (n/2)(a1 + an).
- Geometric sequences have each term obtained by multiplying the preceding term by a constant ratio r, and their series are calculated differently depending on if r = 1 or r ≠ 1.
Applied Math 40S June 2 AM, 2008
The document discusses arithmetic sequences, including defining them recursively or through an implicit linear equation, finding the common difference and nth term, and examples of determining terms and sums of arithmetic sequences given initial terms or the values of a and d. It also provides homework problems involving identifying arithmetic sequences, writing terms, and calculating sums of sequences.
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 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.
L9 sequences and series
The document defines sequences, arithmetic sequences, geometric sequences, and harmonic sequences. It provides examples of calculating terms, finding sums of finite sequences, and determining infinite geometric series sums. Key points covered include the definitions of common difference, common ratio, and nth terms of different sequence types as well as examples of solving problems involving sequence terms and sums.
Chapter 3 sequence and series
The document defines sequences and series, and describes three types of progressions: arithmetic, geometric, and harmonic. It then focuses on arithmetic progressions (AP), defining them as sequences where the difference between consecutive terms is constant. The nth term of an AP is given as an + (n-1)d, where a is the first term and d is the common difference. The sum of the first n terms of an AP is provided as (2a + (n-1)d)/2. Arithmetic means are also discussed, along with finding the number of arithmetic means between two numbers in an AP.
Chapter 3 sequence and series
The document defines sequences and series, and describes three types of progressions: arithmetic, geometric, and harmonic. It then focuses on arithmetic progressions (AP), defining them as sequences where the difference between consecutive terms is constant. The nth term of an AP is given as an + (n-1)d, where a is the first term and d is the common difference. The sum of the first n terms of an AP is provided as (2a + (n-1)d)/2. Arithmetic means are also discussed, along with finding the number of arithmetic means between two numbers in an AP.
Series in Discrete Structure || Computer Science
The document defines and provides examples of sequences, series, arithmetic progressions, and geometric progressions. It gives the formal definitions and notations used to represent these concepts. Several examples are worked through to demonstrate finding explicit formulas for sequences based on given terms, computing specific terms of sequences, and identifying terms of sequences based on given properties like the first term, common difference or ratio, and a target term value.
MATHS PPT 2 (2).pptx
A sequence is an ordered list of numbers where each subsequent term is obtained by applying a specific rule to the previous term(s). A sequence can be finite, with a limited number of terms, or infinite and continue indefinitely. A series is the sum of the terms in a sequence, denoted by Σan where an is the nth term. There are different types of series such as arithmetic where a constant is added to each term, geometric where a constant is multiplied to each term, and harmonic where the reciprocal of each term is added. An arithmetic progression is a specific type of sequence where each term is obtained by adding a fixed constant, called the common difference, to the preceding term.
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Arithmetic and geometric_sequences
- 1. REVIEW Create a line graph for the following interval ( -6, 3] or [7, - 11) ARITHMETIC AND GEOMETRIC SEQUENCES
- 3. SEQUENCE A sequence is an ordered list of numbers that often form a pattern. Each number in the list is called a term of sequence. Describe the pattern in the following sequences and write the next two terms a.) 5,11,17,23 c.) 400, 200, 100, 50 b.) 2, -4, 8, -16 d.) -15, -11, -7, -3 ARITHMETIC AND GEOMETRIC SEQUENCES
- 4. ARITHMETIC SEQUENCE In an arithmetic sequence, the difference between consecutive terms is constants. This difference is called the common difference. Example: 3, 8, 13, 18 ARITHMETIC AND GEOMETRIC SEQUENCES
- 5. ARITHMETIC SEQUENCE Tell whether the sequence is arithmetic. If it is, what is the common difference? a.) 8, 15, 22, 30 b.) 7, 9, 11, 13 c.) 10, 4, -2, -8 d.) 2, -2, 2, -2 ARITHMETIC AND GEOMETRIC SEQUENCES
- 6. RECURSIVE AND EXPLICIT FORMULA Let n= the term number in the sequence Let A(n) = the value of the nth term of the sequence Let d= the common difference Recursive Formula Explicit Formula A(n) = A( n – 1) + d A(n) = A(1) + (n – 1)d ARITHMETIC AND GEOMETRIC SEQUENCES
- 7. RECURSIVE AND EXPLICIT FORMULA Example: An arithmetic sequence is represented by the recursive formula A(n) = A (n – 1) + 12. If the first sequence is 19, write the explicit formula. ARITHMETIC AND GEOMETRIC SEQUENCES
- 8. RECURSIVE AND EXPLICIT FORMULA An arithmetic sequence is represented by the explicit formula A(n) = 32 + (n – 1) (22). What is the recursive formula? ARITHMETIC AND GEOMETRIC SEQUENCES
- 9. RECURSIVE AND EXPLICIT FORMULA 1.) Write a recursive formula for the sequence: 99, 88, 77, 66 2.) Write a recursive formula for the explicit formula: A(n) = 5 + (n -1) (3) 3.) Find the 2nd , 4th , and 11th term: A(n) = - 3+ (n – 1) (5) ARITHMETIC AND GEOMETRIC SEQUENCES
- 10. RECURSIVE AND EXPLICIT FORMULA Write a recursive and an explicit formula for the arithmetic sequence: 9, 7, 5, 3, 1 Find the 7th and 11th term ARITHMETIC AND GEOMETRIC SEQUENCES
- 11. RECURSIVE AND EXPLICIT FORMULA Find the second, third, and fourth terms of the sequence. Then write the explicit formula that represents the sequence A(n) = A (n – 1) – 1 ; A(1) = 8 ARITHMETIC AND GEOMETRIC SEQUENCES