Questions and Answers
What are sequences investigated with?
Sequence inputs/n-values can only be?
0 or positive integers
What are arithmetic sequences?
Sequences generated by adding a constant to the previous term.
What are geometric sequences?
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In sequence equations, what is 'n'?
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What symbols are used to write sequence equations?
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What is t(0)?
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What is t(1)?
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What is the growth rate/common difference (d) of arithmetic sequences?
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What type of graph do linear functions produce?
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What type of graph do arithmetic sequences produce?
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What is the explicit equation for arithmetic sequences?
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What is the explicit equation for the arithmetic sequence -5, -3, -1, 1, 3?
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What is the recursive equation for arithmetic sequences?
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What is the recursive equation for the arithmetic sequence 8, 5, 2, -1, -4?
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For recursive equations, what is needed?
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What is the representation of a recursive equation for an arithmetic sequence?
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What is the multiplier/common ratio (r) of geometric sequences?
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What type of graph do geometric sequences produce?
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What is the explicit equation for geometric sequences?
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What is the explicit equation for the geometric sequence 8, 32, 128, 512, 2048?
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What is the recursive equation for geometric sequences?
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What is the recursive equation for the geometric sequence 8, 32, 128, 512, 2048?
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Study Notes
General Concepts
- SEQUENCES consist of patterns and rules governing the arrangement of numbers.
- Sequence inputs, or n-values, are restricted to non-negative integers.
Types of Sequences
- Arithmetic Sequences: Created by adding a constant value (common difference) to the previous term.
- Geometric Sequences: Formed by multiplying the previous term by a constant value (common ratio).
Sequence Variables and Terms
- In sequence equations, "n" refers to the position of a term in the sequence.
- Different notations used for sequences include t(n), tn, t, a(n), t(n+1), and t(n-1).
- t(0) identifies the 0th term, while t(1) refers to the 1st term.
Arithmetic Sequences
- The GROWTH RATE or COMMON DIFFERENCE (d) in arithmetic sequences is constant and analogous to the slope of the line.
- GRAPHS of arithmetic sequences depict a straight line but are discrete (not continuous).
- Explicit Equation: For arithmetic sequences, the formula is t(n) = d(n) + t(0).
- Example Explicit Equation: For the sequence -5, -3, -1, 1, 3, it is expressed as t(n) = 2(n) - 7.
- Recursive Equation: Defined as t(n+1) = t(n) + d, requiring an initial term such as t(0) or t(1).
Geometric Sequences
- The MULTIPLIER or COMMON RATIO (r) signifies the factor by which each term is multiplied to obtain the next term.
- GRAPHS of geometric sequences illustrate a curved exponential line and are also discrete.
- Explicit Equation: For geometric sequences, the equation is t(n) = t(0) (r)ⁿ.
- Example Explicit Equation: For the sequence 8, 32, 128, 512, 2048, the equation is t(n) = 2(4)ⁿ.
- Recursive Equation: Expressed as t(n+1) = r * t(n), necessitating initial terms t(0) and t(1).
Examples of Sequences
- Arithmetic Recursive Equation for 8, 5, 2, -1, -4: t(n+1) = t(n) - 3 with d = -3, starting from t(1) = 8.
- Geometric Recursive Equation for 8, 32, 128, 512, 2048: t(n+1) = 4 * t(n) with t(0) = 2 and t(1) = 8.
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Description
Explore the fundamentals of sequences in Algebra 2 with this set of flashcards. Learn about arithmetic and geometric sequences, including their definitions and rules. Perfect for mastering the topic and preparing for exams.