Algebra 2 Sequences Flashcards

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?

Uses the Greek letter 'sigma' to denote the sum of a sequence defined by a rule. Signup and view all the answers

What is the summation formula for constant series?

nc ex. (3+3+3+3+3.....) Signup and view all the answers

What is the summation formula for linear series?

n(n+1)/2 ex. (1+2+3+4+5+6.....) Signup and view all the answers

What is the summation formula for quadratic series?

n(n+1)(2n+1)/6 ex. (2+4+8+16+32.....) Signup and view all the answers

What is the general rule for arithmetic sequences?

An = A1 + (n-1)d Signup and view all the answers

What does Sn represent in the context of arithmetic series?

Sn = n(A1 + An) / 2 Signup and view all the answers

What characterizes a geometric sequence?

The ratio of successive terms is a constant called the common ratio r (r is not equal to 1). Signup and view all the answers

What is the general rule for geometric sequences?

An = A1r^(n-1) Signup and view all the answers

What is the geometric mean of two numbers a and b?

Square root of ab. Signup and view all the answers

What is the sum of the first n terms of a geometric series?

Sn = A1(1 - r^n) / (1 - r), r is not equal to 1. Signup and view all the answers

What defines an infinite geometric series?

A geometric series that has infinitely many terms. Signup and view all the answers

What does it mean to converge in the context of series?

When |r| < 1 and the partial sum approaches a fixed number. Signup and view all the answers

What is a limit in mathematical analysis?

The number that the partial sums approach as n increases. Signup and view all the answers

What does it mean to diverge in series?

When |r| is greater than or equal to 1 and the partial sum does not approach a fixed number. Signup and view all the answers

What is the formula for the sum of an infinite geometric series?

S = A1 / (1 - r) Signup and view all the answers

What is proof by mathematical induction?

<ol> <li>The base case: show that the statement is true for n=1; 2. Assume that the statement is true for a natural number k; 3. Prove that the statement is true for the natural number k+1.</li> </ol> Signup and view all the answers

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

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 ).

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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.