Notes on Sequences and Series Lecture
Introduction
- Importance of mathematics in daily life.
- Applications in civil engineering, economics, investment, software development, art, and music.
Topics Covered
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Sequences
- Definition: An arrangement of numbers in a specific order according to a rule.
- Examples of types include:
- Arithmetic Sequences: A sequence where the difference between consecutive terms is constant.
- Geometric Sequences: A sequence where the ratio of consecutive terms is constant.
- Harmonic Sequences: Related to sequences formed by the reciprocals of an arithmetic sequence.
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Types of Sequences and Series
- Sequence: The ordered list of numbers.
- Series: The sum of the terms of a sequence.
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Formulas for Different Sequences
- Formula for arithmetic sequence:
- General term: ( a_n = a_1 + (n-1)d )
- Where ( d ) is the common difference.
- Formula for geometric sequence:
- General term: ( a_n = a_1 imes r^{(n-1)} )
- Where ( r ) is the common ratio.
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Recursive Sequences
- Defined using previous terms, e.g. Fibonacci Sequence: ( F_n = F_{n-1} + F_{n-2} )
- Importance of understanding both definitions and formulas.
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Examples and Activities
- Find the next two terms in various sequences.
- Activity display patterns in sequences (e.g. number of people on building floors).
- Examination of relationships between terms.
- Graphing sequences, beginning with simple formulas like ( a_n = 2n + 1 ).
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Fibonacci Sequence
- Notable for its presence in nature and pattern of growth.
- Each term is the sum of the two preceding terms, illustrating its unique recursive nature.
- Applications in modeling and biology (e.g., rabbit population growth).
Conclusion
- Key takeaways on the importance of understanding sequences and series.
- Reinforcement of mathematical concepts through applications in real-world contexts.
Mathematical Properties and Definitions
- Definition of terms and notation for sequences.
- Distinction between finite and infinite sequences.
- Clear notation and visual representation for understanding complex ideas clearly.
Note: Each section of these notes expands on concepts covered in class and may serve as a reference for future assignments and exercises.