change in arithmetic and geometric sequences

Arithmetic explicit formula

An=Ak+d(n-k)

Geometric Explicit Formula

An= A1r^(n-1)

Arithmetic Sequences

Have the same differences between terms (add)

Geometric Sequence

Any sequence with a constant ratio between consecutive terms

What do arithmetic sequences relate to?

Linear Functions

y=mx+b describes a continuous function.

Domain of arithmetic sequences

Restricted to natural numbers/positive integers but will never be all real numbers because sequences are defined for discrete, specific positions

Range of arithmetic sequences

Depends on common difference.

If d>0, the terms increase and the range could be infinite in a positive direction.

If d<0, terms decrease and range could be infinite in a negative direction

Difference between sequence and function

Sequence: List of numbers in a specific order with whole numbers as their position; clear pattern and position

Function: a rule that works with any number, not just whole numbers, and has a wide range of input values; relationship between input and output values

What do geometric sequences relate to?

exponential function

f(x)=a*b^x

a= initial value (cannot equal 0 and is a real number)

b= positive real number (b>0) and growth factor

x= exponent

Domain for exponential function

all real numbers

Characteristics of Exponential Functions

a>0 or b>1 = exponential growth

a>0 and 0<b<1= exponential decay

End behavior of exponential function

if b>1, limit x to positive infinity will be positive infinity and limit x to negative infinity will be 0

if b<1, limit x to positive infinity will be 0 and limit x to negative infinity will be positive infinity

f(x)=b^x

Domain: all real numbers

Range: positive real numbers

b>1, increasing function, continuous, and concave up

0<b<1, decreasing function, continous, and concave up