Questions and Answers
What is a recursive formula?
A rule in which one or more previous terms are used to generate the next term.
What is an explicit formula?
Defines the nth term of a sequence as a function of n.
What is a partial sum?
Indicated by Sn, it is the sum of a specified number of terms of a sequence.
What does summation notation represent?
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What is the summation formula for constant series?
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What is the summation formula for linear series?
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What is the summation formula for quadratic series?
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What is the general rule for arithmetic sequences?
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What does Sn represent in the context of arithmetic series?
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What characterizes a geometric sequence?
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What is the general rule for geometric sequences?
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What is the geometric mean of two numbers a and b?
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What is the sum of the first n terms of a geometric series?
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What defines an infinite geometric series?
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What does it mean to converge in the context of series?
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What is a limit in mathematical analysis?
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What does it mean to diverge in series?
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What is the formula for the sum of an infinite geometric series?
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What is proof by mathematical induction?
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Study Notes
Recursive and Explicit Formulas
- Recursive Formula: Generates each term using one or more preceding terms.
- Explicit Formula: Provides a direct formula for the nth term based on its position.
Summation Concepts
- Partial Sum (Sn): Represents the sum of a designated number of terms in a sequence.
- Summation Notation: Utilizes the Greek letter "sigma" (Σ) to indicate the sum defined by a rule.
Summation Formulas
- Constant Series: The sum of n terms from a constant series can be represented as ( S_n = n \cdot c ), e.g., ( 3 + 3 + 3 + \ldots ).
- Linear Series: Sum of first n integers is given by ( S_n = \frac{n(n + 1)}{2} ), representing sequences like ( 1 + 2 + 3 + \ldots ).
- Quadratic Series: Sum formula is expressed as ( S_n = \frac{n(n + 1)(2n + 1)}{6} ), for sequences involving squares.
Arithmetic Sequences
- General Rule: The nth term is calculated by ( A_n = A_1 + (n-1)d ), where ( d ) is the common difference.
- Sum of First n Terms: The sum is determined by ( S_n = \frac{n(A_1 + A_n)}{2} ).
Geometric Sequences
- Definition: A sequence where each term is multiplied by a constant called the common ratio ( r ) (with ( r \neq 1 )).
- General Rule: The nth term is given by ( A_n = A_1 r^{n-1} ).
- Geometric Mean: Calculated as the square root of the product of two numbers ( ab ).
Geometric Series
- Sum of First n Terms: For a geometric series, the sum of the first n terms is given by ( S_n = \frac{A_1(1 - r^n)}{1 - r} ), where ( r \neq 1 ).
- Infinite Geometric Series: Contains infinitely many terms and converges only when ( |r| < 1 ).
- Convergence: When a series approaches a fixed number as partial sums increase.
- Divergence: Occurs when ( |r| \geq 1 ), with no convergence to a fixed sum.
- Sum of Infinite Series: For converging series, the sum can be expressed as ( S = \frac{A_1}{1 - r} ).
Mathematical Induction
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Proof Process:
- Establish the statement holds for the base case ( n = 1 ).
- Assume the statement is true for ( k ).
- Demonstrate it remains true for ( k + 1 ).
Studying That Suits You
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Description
Test your knowledge of key concepts in sequences with these flashcards tailored for Algebra 2. Each card presents important terms like recursive and explicit formulas as well as summation notation and partial sums. Perfect for quick revision or deeper understanding of algebraic sequences.