Algebra 2 - Sequences Flashcards

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Questions and Answers

What are sequences investigated with?

Numbers

Rules

Patterns

Patterns and rules (correct)

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?

Sequences generated by multiplying the previous term by a constant. Signup and view all the answers

In sequence equations, what is 'n'?

Number of term Signup and view all the answers

What symbols are used to write sequence equations?

t(n), tn, t, a(n), t(n+1), t(n-1) Signup and view all the answers

What is t(0)?

The 0th term Signup and view all the answers

What is t(1)?

The 1st term Signup and view all the answers

What is the growth rate/common difference (d) of arithmetic sequences?

Also the slope and are constant Signup and view all the answers

What type of graph do linear functions produce?

Straight linear line, continuous Signup and view all the answers

What type of graph do arithmetic sequences produce?

Straight linear line, disconnected/discrete Signup and view all the answers

What is the explicit equation for arithmetic sequences?

t(n) = d(n) + t(0) Signup and view all the answers

What is the explicit equation for the arithmetic sequence -5, -3, -1, 1, 3?

t(n) = 2(n) - 7 Signup and view all the answers

What is the recursive equation for arithmetic sequences?

t(n+1) = t(n) + d Signup and view all the answers

What is the recursive equation for the arithmetic sequence 8, 5, 2, -1, -4?

t(n+1) = t(n) - 3 Signup and view all the answers

For recursive equations, what is needed?

A starting/initial point Signup and view all the answers

What is the representation of a recursive equation for an arithmetic sequence?

If t(100) = 193 then t(101) = t(100) + 2 Signup and view all the answers

What is the multiplier/common ratio (r) of geometric sequences?

The multiplied value from each y-value for every consecutive x-value Signup and view all the answers

What type of graph do geometric sequences produce?

Curved exponential line, discrete/disconnected Signup and view all the answers

What is the explicit equation for geometric sequences?

t(n) = t(0) (r)ⁿ Signup and view all the answers

What is the explicit equation for the geometric sequence 8, 32, 128, 512, 2048?

t(n) = 2(4)ⁿ Signup and view all the answers

What is the recursive equation for geometric sequences?

t(n+1) = r * t(n) Signup and view all the answers

What is the recursive equation for the geometric sequence 8, 32, 128, 512, 2048?

t(n+1) = 4 * t(n) Signup and view all the answers

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.