•
•
To explore exponential growth and decay
To discover the connection between recursive and
exponential forms of geometric sequences
Recursive Routines
• Recursive routines are useful for seeing how a
sequence develops and for generating the first
few terms.
• But if you’re looking for the 50th term, you’ll
have to do many calculations to find your
answer.
• In chapter 3, you found that graphs of the points
formed a linear pattern, so you learned to write
the equation of a line.
Recursive Routines
• Recursive routines with a constant multiplier
create a different pattern. In this lesson you’ll
discover the connection between these recursive
routines and exponents.
• Then with a new type of equation you’ll be able
to find any term in a sequence based on a
constant multiplier without having to find all the
terms before it.
Loosing Area in a Square Fractal
• In this investigation you will look for patterns in
area of a square fractal.
Growth of a
Rectangular
Fractal
• To create a
fractal we will
begin with a
27 x 27
square
• This is stage
0.
• To create
stage 1 draw
two vertical
lines and two
horizontal
lines to
subdivide the
shape into 9
equal parts.
Shade any
one of the
parts to
illustrate the
square being
removed.
• To create stage
2 draw two
vertical lines
and two
horizontal lines
in each of the
remaining
squares to
subdivide the
remaining
squares into 9
equal parts
each. Shade
the same one
part of each of
these squares to
represent them
being removed.
• To create stage 3
draw two vertical
lines and two
horizontal lines in
each of the
remaining squares
to subdivide the
square into 9 equal
parts. Shade the
same one part of
each of these
squares to
represent the
square being
removed.
Stage
Number
0
Total
Unshaded
Area
Ratio of this
Stage’s area to
the previous
Stage’s area
729
1
648
2
576
3
512
Let’s collect some data
Stage
Number
0
Total
Unshaded
Area
Ratio of this
Stage’s area to
the previous
Stage’s area
729
1
648
2
576
3
512
648 8

729 9
576 8

648 9
512 8

576 9
Let’s collect some data
Stage
Number
0
Total
Unshaded
Area
Ratio of this
Stage’s area to
the previous
Stage’s area
729
1
648
2
576
3
512
648 8

729 9
576 8

648 9
512 8

576 9
1
8
1
455
4
512g  455
9 8
9
9
512
9
Use the ratio to predict the area of stage 4
Stage
Number
0
Total
Unshaded
Area
Ratio of this
Stage’s area to
the previous
Stage’s area
729
8
648 8
729g
1

9
729 9
2
576 8
8


2

729g 
648 9
 9 3
512 8
8

3
729g 
576 9
9
4
1
455
8
4
9 8
729g 
9
512
9

Rewrite each total unshaded area using the constant
multiplier.
Stage
Number
0
1
2
3
4
Total
Unshaded
Area
729
8
729g
9
2
8
729g 
 9 3
8
729g 
9
4
8
729g 
9
If x is the stage number
write an expression for
the unshaded area in
stage x.
8
y  729  
9
x
Stage
Number
0
1
2
3
4
Total
Unshaded
Area
729
8
729g
9
2
8
729g 
 9 3
8
729g 
9
4
8
729g 
9
8
y  729  
9
x
Create a graph for this equation.
Check the calculator table to
see that it contains the same
values as your table.
What does the graph tell you
about the area of the
rectangular fractal.
Total length
4
y  27  
3
x
Stage
Number
Constant
multiplier
Starting
length
This type of equation is called an exponential equation.
The standard form of an exponential
equation is
y  ab
x
3
4
y  27  
3
Exponential Form
 4 4 4
y  27      
33 3
Expanded Form
Example
• Seth deposits $200 in a savings account. The
account pays 5% annual interest. Assuming that
he makes no more deposits and no withdrawals,
calculate his new balance after 10 years.
• Determine the constant multiplier.
• Write an equation that can be used to calculate
the yearly total.
Time
Expanded Form
Starting
200
After 1 yr.
200(1+0.05)
200(1+0.05)1
210
After 2
yrs.
200(1+0.05)(1+0.05)
200(1+0.05)2
220.50
After 3
yrs.
200(1+0.05)(1+0.05)(1+0.05)
200(1+0.05)3
231.53
After x
yrs.
200(1+0.05)(1+0.05)…(1+0.05)
200(1+0.05)x
After 10 years
Exponential
Form
New
Balance
200
y  200(1  0.05)10