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Questions and Answers
Match the following characteristics with the correct sequence type:
Constant difference between terms = Arithmetic sequence Common ratio between terms = Geometric sequence Formula for the nth term: $a_n = a + (n-1)d$ = Arithmetic sequence Formula for the nth term: $a_n = a imes r^{(n-1)}$ = Geometric sequence
Match the following properties with the corresponding sequence type:
Addition or subtraction to get consecutive terms = Arithmetic sequence Multiplication or division to get consecutive terms = Geometric sequence Example: $2, 4, 6, 8, ...$ = Arithmetic sequence Example: $3, 6, 12, 24, ...$ = Geometric sequence
Match the following descriptions with the appropriate type of sequence:
The ratio of any term to its preceding term is constant = Geometric sequence The difference between any two consecutive terms is constant = Arithmetic sequence Each term is obtained by multiplying the previous term by a fixed number = Geometric sequence Each term is obtained by adding a fixed number to the previous term = Arithmetic sequence
Match the following characteristics with the correct type of sequence:
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Match the following formulas with the corresponding sequence type:
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Match the following growth patterns with the appropriate type of sequence:
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Match the following terms with the correct type of sequence:
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Study Notes
Sequence Types Overview
- Different types of sequences include arithmetic, geometric, Fibonacci, and harmonic sequences, each with unique characteristics.
Arithmetic Sequence
- Defined by a constant difference between consecutive terms.
- General formula: ( a_n = a_1 + (n-1)d ) where ( d ) is the common difference.
- Growth pattern is linear, resulting in a straight line when graphed.
Geometric Sequence
- Defined by a constant ratio between consecutive terms.
- General formula: ( a_n = a_1 \cdot r^{(n-1)} ) where ( r ) is the common ratio.
- Growth pattern is exponential, leading to a curve that increases rapidly when ( r > 1 ).
Fibonacci Sequence
- Each term is the sum of the two preceding terms, starting from 0 and 1.
- The sequence progresses as follows: 0, 1, 1, 2, 3, 5, 8, ...
- Growth pattern approximates the Golden Ratio as it progresses.
Harmonic Sequence
- Formed by taking the reciprocals of an arithmetic sequence.
- General formula: ( a_n = \frac{1}{a_1 + (n-1)d} ).
- Growth pattern decreases and approaches zero but never reaches it, creating a curve that flattens as ( n ) increases.
Matching Characteristics
- Identify specific characteristics of each type to match to their sequence type:
- Constant difference suggests an arithmetic sequence.
- Constant ratio indicates a geometric sequence.
- Sum of two previous terms points to the Fibonacci sequence.
- Reciprocals of an arithmetic sequence define a harmonic sequence.
Applications and Context
- Sequences appear in various real-world applications such as finance (compound interest for geometric sequences), nature (Fibonacci in populations), and computer science (algorithms).
- Understanding properties helps to determine the type of sequence, aiding in solving mathematical problems and analyzing data trends effectively.
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Description
Test your understanding of geometric and arithmetic sequences by matching their characteristics with the correct sequence type. Identify the properties that distinguish these two fundamental types of sequences.
