Superlesson
Project 4-2
Advanced Algebra
Chapter 4, Patterns and Structure in Algebra
Geometric sequences and series have a variety of applications, from cell division to consumer fraud to the problem of whether a snail can complete a long and perilous journey!
Part A, Geometric Sequences
1. Read about mitosis, the most common type of cell division in plant and animal cells.
a. Complete the table to show how many cells exist after each of the first ten divisions.
| divisions | 0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
| cells | 1 |
b. Do the values in Row 2 of the table form a geometric sequence? Explain.
c. Write an expression the 100 th term of the sequence.
Part B, Geometric Series
2. "Pyramid schemes" are illegal for good reason: they claim that people who join the scheme will make large amounts of money, when in fact those people may well lose money. Read about pyramid schemes.
a. One common pyramid scheme is a chain letter. Read Mr. Smith's letter below. If Mr. Smith follows the directions on the chain letter, and if all the people on the list follow the directions, how much money will Mr. Smith receive after the first round of the letter?
Hello Mr. Smith!
Below is a list of 5 people. Send $1 to each person on the list. Then cross the first name off the list, add your name to the bottom of the list, and send the letter to 10 of your friends.
1. Mr. Henson, 111 Main St., Chicago IL 26519
2. Mrs. Jones, 5 Maple Lane, Finnegan OH 75123
3. Mr. Mason, 44 Oak St., Beatty, UT 33221
4. Mrs. Klein, 502 Maple, Erie, PA 77091
5. Mr. Webb, 51 Pine St., Detroit, MI 98765
b. If everyone continues to follow directions, what is the total amount of money Mr. Smith will have received after the second round? after the third round?
c. Suppose that everyone continues to follow directions. Write a geometric series expressing the total amount Mr. Smith will have received after the n th round.
d. Find the total amount Mr. Smith will have received after the 8 th round.
e. Why, in all likelihood, will Mr. Smith not receive the amount you calculated above?
Part C, Investigating the Concept of a Limit
4. Read about Zeno's Paradoxes.
a. A snail set out to travel a distance of 1 yard. In the first stage of its journey it traveled 1/2 yard. In Stage 2 it traveled half the remaining distance. It continued in this fashion, at each stage traveling half the remaining distance. Write a geometric series giving the distances the snail traveled during each of the first five stages of the journey.
b. What is the common ratio of the series?
c. What is the sum of the infinite series which has the above terms as its first five terms? Is the sum equal to the distance the snail must travel?
d. Explain Zeno's Paradox in relation to this problem.