[size=150]For the function [math]f(x)=32\cdot(\frac{3}{4})^x[/math], evaluate mentally:[br][/size][math]f(0)[/math]
[size=150]The first person who comes in takes [math]\frac{1}{3}[/math] of the cake. Then a second person takes [math]\frac{1}{3}[/math] of what is left. Then a third person takes [math]\frac{1}{3}[/math] of what is left. And so on.[/size]
Write two definitions for [math]C[/math]: one recursive and one non-recursive.[br]
What is a reasonable domain for this function? Be prepared to explain your reasoning.[br]
[size=150]Draw another square of side length 1 that shares a side with the first square. Next, add a 2-by-2 square, with one side along the sides of [i]both[/i] of the first two squares. Next, add a square with one side that goes along the sides of the previous two squares you created. Next, do it again.[br]Pause here for your teacher to check your work.[/size]
Write a sequence that lists the side lengths of the squares you drew.[br]
Predict the next two terms in the sequence and draw the corresponding squares to check your predictions.[br]
Describe how each square’s side length depends on previous side lengths.[br]
Let [math]F(n)[/math] be the side length of the [math]n^{th}[/math] square. So [math]F(1)=1[/math] and [math]F(2)=1[/math]. Write a recursive definition for [math]F[/math].[br]
Is the Fibonacci sequence arithmetic, geometric, or neither? Explain how you know.
[size=150]Look at quotients [math]\frac{F(1)}{F(0)},\frac{F(2)}{F(1)},\frac{F(3)}{F(2)},\frac{F(4)}{F(3)},\frac{F(5)}{F(4)}[/math]. [/size][br]What do you notice about this sequence of numbers?
The 15th through 19th Fibonacci numbers are 610, 987, 1597, 2584, 4181. [br]What do you notice about the quotients[math]\frac{F(16)}{F(15)},\frac{F(17)}{F(16)},\frac{F(18)}{F(17)},\frac{F(19)}{F(18)}[/math]?[br]