Algebra Formulas: Sequences & Functions

Questions and Answers

What is the recursive formula for geometric sequences?

a sub 1 = first term, a sub n = r(a sub n-1)

What is the explicit formula for geometric sequences?

a sub n = a sub 1 (r^n-1)

What is the formula for exponential growth?

y = a (1 + r)^t

What is the formula for exponential decay?

y = a (1 - r)^t Signup and view all the answers

What is the recursive formula for arithmetic sequences?

a sub 1 = first term, a sub n = a sub n-1 + d Signup and view all the answers

What is the explicit formula for arithmetic sequences?

a sub n = a sub 1 + d (n-1) Signup and view all the answers

What is the general formula for an exponential function?

y = ab^x Signup and view all the answers

What is the domain and range for exponential functions?

Domain = all real numbers, Range = y > 0/all real positive numbers Signup and view all the answers

Study Notes

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 ).

Description

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.