Volume of Fractal POP-UP
Task: Our task is to find the volume of the fractal pop-up constructed in class. We will be
applying techniques used in class that involve sequences, series and recursion.
You must answer each question and show work in the space provided.
In class we constructed 3 stages (iterations) of the pop-up. The initial square was 8 cm by 8 cm.
Find the volume of the rectangular prism formed in stage 1, call it V1. Is it a power of 2? Express
it as such.
For the second stage of the pop-up, 2 more rectangular prisms were formed. Find the volume of
one of them, and then write the expression for the total volume at this stage. Write it recursively,
and then simplify it by writing your answer using exponents.
V2= V1 + _________ = ___________
Follow the same procedure and write the volume for the third stage of the pop-up.
V3= V2 + _________ = ___________
Do you see a pattern? Describe it.
Use it. Write an expression using recursion for the volume at the 6th stage of the pop-up.
V6 = V_ + _________
Write the expression explicitly by just using exponents.
V6 = ______________________________
What do you think the volume will be at the 10th stage? 20th , 100th?
Can you identify the pattern between the subscript and the exponent of the last term? I wrote it in
the HINT for you below, verify that it is correct.
Vn = 2 7 25 23 21 211 213 21
M where M = -2n+9.
What type of series do you see in each of the volumes and why? ____________________
What is the first term?_____________ What is the common ratio?___________
Write the series
Series: ______________________ Sum: __to be determined_________________
The REAL question:
Suppose this fractal pop-up process never ends and we are asked to find the volume of the
pop-up anyway…. Can we do it? Do you think the volume will just continue to increase or
will it converge to some value. In other words, if this were a gift for someone would you be
able to find a box big enough to pack it in?
Write the series using sigma notation and calculate the sum. Use your mathematics and
confirm with technology. BC students- use your knowledge to determine if the series will
converge. No calculator for you.
Show ALL work in the space below.