Define and give examples of arithmetic sequence, geometric sequence, and Fibonacci sequence.
Understand the Problem
The question lists different types of sequences and series, specifically arithmetic sequences, geometric sequences, and Fibonacci sequences. It may be asking for definitions, properties, or examples of these mathematical concepts.
Answer
Arithmetic sequence: common difference; Geometric sequence: common ratio; Fibonacci sequence: sum of two previous terms.
Answer for screen readers
An arithmetic sequence is defined by a constant difference between terms, a geometric sequence has a constant ratio, and a Fibonacci sequence is defined by each term being the sum of the two preceding terms.
Steps to Solve
- Define Arithmetic Sequence
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the "common difference".
For example, in the arithmetic sequence (2, 4, 6, 8, 10), the common difference is (2).
- Define Geometric Sequence
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the "common ratio".
For example, in the geometric sequence (3, 6, 12, 24), the common ratio is (2) because (6 = 3 \times 2), (12 = 6 \times 2), and so on.
- Define Fibonacci Sequence
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with (0) and (1).
The Fibonacci sequence starts like this: (0, 1, 1, 2, 3, 5, 8, 13).
- List Properties of Each Sequence
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Arithmetic Sequence Properties:
- The (n)-th term can be expressed as: $$ a_n = a_1 + (n-1)d $$ where (a_1) is the first term and (d) is the common difference.
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Geometric Sequence Properties:
- The (n)-th term can be expressed as: $$ a_n = a_1 \cdot r^{(n-1)} $$ where (a_1) is the first term and (r) is the common ratio.
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Fibonacci Sequence Properties:
- The (n)-th term can be found using: $$ F_n = F_{n-1} + F_{n-2} $$ with initial conditions (F_0 = 0) and (F_1 = 1).
An arithmetic sequence is defined by a constant difference between terms, a geometric sequence has a constant ratio, and a Fibonacci sequence is defined by each term being the sum of the two preceding terms.
More Information
Each of these sequences has its own unique properties and applications. Arithmetic sequences are often seen in daily life situations like calculating total costs, while geometric sequences can model exponential growth, like population growth or compound interest. The Fibonacci sequence appears in nature, like in the arrangement of leaves and the branching of trees.
Tips
- Confusing arithmetic and geometric sequences, particularly in identifying common difference versus common ratio.
- Miscalculating terms in the Fibonacci sequence by not remembering to sum the two previous numbers.
- Forgetting initial conditions when dealing with Fibonacci sequences, which start with (F_0 = 0) and (F_1 = 1).
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