Algebra Formulas: Sequences & Functions

What is the recursive formula for geometric sequences?

What is the explicit formula for geometric sequences?

What is the formula for exponential growth?

What is the formula for exponential decay?

What is the recursive formula for arithmetic sequences?

What is the explicit formula for arithmetic sequences?

What is the general formula for an exponential function?

What is the domain and range for exponential functions?

Geometric Sequences

Recursive Formula:
- First term denoted as ( a_1 ).
- Subsequent terms calculated as ( a_n = r(a_{n-1}) ), where ( r ) is the common ratio.
Explicit Formula:
- General formula given by ( a_n = a_1 (r^{n-1}) ).

Exponential Functions

Exponential Growth Formula:
- Represents growth over time, formulated as ( y = a(1 + r)^t ).
- ( a ) is the initial amount, ( r ) is the growth rate, and ( t ) is time.
Exponential Decay Formula:
- Describes decrease over time using ( y = a(1 - r)^t ).
- Similar parameters apply: ( a ) is the initial amount, ( r ) is the decay rate, and ( t ) is time.

Arithmetic Sequences

Recursive Formula:
- First term defined as ( a_1 ).
- Each term calculated with ( a_n = a_{n-1} + d ), where ( d ) is the common difference.
Explicit Formula:
- General term expressed as ( a_n = a_1 + d(n - 1) ).

Exponential Functions Characteristics

Function Representation:
- An exponential function is represented by the equation ( y = ab^x ).
Domain and Range:
- Domain consists of all real numbers.
- Range includes all real positive numbers, expressed as ( y > 0 ).

This quiz features flashcards covering essential algebra formulas related to arithmetic and explicit sequences, as well as exponential functions. Each card includes definitions for key concepts like geometric sequences and exponential growth/decay formulas that are crucial for understanding advanced algebra topics.