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Geometric Sequences definitions

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  • Geometric Sequence
    A list of numbers where each term is found by multiplying the previous term by a fixed value called the common ratio.
  • Arithmetic Sequence
    A list of numbers where each term is found by adding a fixed value, known as the common difference, to the previous term.
  • Common Ratio
    A constant multiplier used to move from one term to the next in a geometric sequence, found by dividing a term by its previous term.
  • Common Difference
    A fixed value added to each term to get the next term in an arithmetic sequence.
  • General Term
    A formula that allows calculation of any term in a sequence based on its position, without listing all previous terms.
  • First Term
    The initial value in a sequence, often denoted as a₁, which serves as the starting point for generating subsequent terms.
  • Exponent
    A mathematical notation indicating how many times a number, such as the common ratio, is multiplied by itself.
  • Sequence
    An ordered list of numbers following a specific pattern or rule.
  • Term
    An individual number or element in a sequence, identified by its position.
  • Position
    The place or index of a term within a sequence, often represented by n.
  • Fraction
    A numerical quantity that is not a whole number, often used in sequences with non-integer common ratios.
  • Power
    The result of raising a number, such as the common ratio, to an exponent, indicating repeated multiplication.
  • Formula
    A mathematical expression that provides a rule for calculating terms in a sequence.
  • aₙ
    A symbol representing the nth term in a sequence, where n indicates the term's position.
  • a₁
    A symbol representing the first term in a sequence, serving as the base for generating other terms.
  • n
    A variable representing the position of a term in a sequence, used in general term formulas.
  • r
    A symbol representing the common ratio in a geometric sequence.
  • n−1
    An expression indicating one less than the term's position, used as the exponent in the general term formula for geometric sequences.
  • Exponential Expression
    A mathematical statement involving a base raised to a power, such as r^(n−1) in geometric sequences.
  • Multiplication
    The operation used to generate each new term in a geometric sequence by applying the common ratio.
  • Division
    The operation used to find the common ratio by comparing consecutive terms in a geometric sequence.
  • General nth Term Formula
    An equation, aₙ = a₁·r^(n−1), that calculates any term in a geometric sequence using the first term and common ratio.
  • Consecutive Terms
    Two terms that appear one after the other in a sequence.
  • Pattern
    A recognizable rule or relationship that defines how terms in a sequence are generated.
  • Series
    The sum of the terms of a sequence, often studied after understanding sequences.