Algebra 2 - Sequences Flashcards

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.