The topic of iteration is essential to the study of dynamical systems. The following handout introduces the concept of iteration at the Algebra I level with suggested extensions for Algebra II. The examples require the use of the TI-82 and include specific instructions for its use. Additional resource problems are also included.
Iteration is the process of using the output of the previous operation as the input for the next operation. In other words, iteration is the process of doing the same thing over and over and over again. The process of iterating can be demonstrated on the graphing calculator in many different ways. The methods used here will be ANS key, lists, sequence mode, and programming. Applications of iteration will follow. Important vocabulary words are italicized.
Prior to iteration two things are needed: a function and a starting value. The starting value is called the seed.
FUNCTION: f(x) = 2x SEED: x = 1
On the TI-82:
STEP 1: go to the Home screen
STEP 2: put in the seed by pressing 1 followed by ENTER
STEP 3: put in the function by pressing 2*ANS followed by ENTER
STEP 4: press ENTER four more times
Record the results starting with the seed value.
______, ______, ______, ______, ______, ______, . . .
The above infinite sequence of numbers is referred to as the orbit of 1. What happens to this orbit as the iterations continue? Continue to press the ENTER key to find out. The values get larger and larger and larger so the fate of the orbit is positive infinity. If the orbit goes to positive or negative infinity, the orbit is said to escape.
Repeat the above process for seed value x = 3 using the same function f(x) = 2x. Record your results: ______, ______, ______, ______, ______. What is the fate of the orbit? Try different seeds: x = .01, -1.4, 0. What is the fate of the orbit in each case? When the fate of the orbit is a constant, that constant is called a fixed point.
Using subscripts, call the seed value xo. Then the value obtained from the first iteration is x1, the value of the second iteration is x2, etc.
x0 = 1
x1 = f(x0) = f(1) = 2
x2 = f(x1) = f(2) = 4
x3 = f(x2) = f(4) = 8
over and over and over again.
The sequence looks like this: x0, x1, x2, x3, x4, x5, . . . , where each term depends on the term before it. To show this dependency, rewrite the sequence using the notation for composition of functions as follows:
x0, f(x0), f(f(x0), f(f(f(x0))), f(f(f(f(x0)))), . . .
Lists can be used to display this information. Lists can also be used to generate this information if the closed form of the sequence is known. Only the display mode will be detailed here.
On the TI-82:
STEP 1: press STAT, press 4:ClrList
STEP 2: after ClrList press L1,L2 followed by ENTER (this clears lists)
STEP 3: press STAT, press 1:Edit...
STEP 4: enter data using L1 for the number of iterations performed and L2 for the value of the function after that number of iterations
STEP 5: press 4:PlotsOff followed by ENTER (turns off all plots)
STEP 6: press STAT PLOT, press 1:Plot1
STEP 7: turn Plot1 ON and make the window look like the following:
TYPE: scatter (first picture)
MARK: + (second picture)
STEP 8: press ZOOM, press 9:ZoomStat
How does this graph relate to the fate of the orbit for xo = 1?
Now go to the Home screen and press L1 followed by ENTER. The list L1 will be displayed horizontally. Press L2 followed by ENTER. Now press L2(4) followed by ENTER. What happens?
[Extensions: Identify the graph of Plot1; write L2 in closed form; generate L2 by setting L2 = closed form; let L3 = log L2 and show (L1,L3) is linear; use regression equations.]
Since each term of the iteration depends on the term in front of it, the function can be rewritten to show this dependency.
function form: f(x) = 2*x
recursion form: xn = 2*xn-1
TI-82 form: Un = 2*Un-1
Then is the recursion formula for 1, 2, 4, 8, 16, . . . Write the recursion formula for 3, 6, 12, 24, 48, . . .
On the TI-82:
STEP 1: press MODE, go down to the fourth line and turn on Seq
STEP 2: press Y=
STEP 3: let Un = 2*Un-1
STEP 4: press TblSet
TblMin = 0 (give starting n value for table)
Tbl = 1 (set increment)
STEP 5: press WINDOW
UnStart = 1 (sets seed)
VnStart = 0 (not in use at this time)
nStart = 0 (where table will start)
nMin = 1 (where graph will start)
nMax = 5 (where graph will end)
Xmin = 0 (sets window size)
Xmax = 8 "
Xscl = 1
Ymin = 0 (sets window size)
Ymax = 256 "
Yscl = 50
STEP 6: press TABLE, scroll down
STEP 7: press GRAPH
Notice that although Un continues to iterate, the graph only shows x1 to x5. The nMin and nMax values control what appears on the graph. Return to WINDOW and change nMax to 8 and GRAPH again.
NOTE: If you would like to see the TABLE and GRAPH simultaneously, go to MODE and change from full screen to split screen.
[Extensions: display iterations using the web diagrams; find fixed points analytically.]
Another way to demonstrate the iteration of the function f(x) = 2x is to create a program.On the TI-82:
STEP 1: press PRGM, over to NEW, followed by ENTER
STEP 2: enter a name: ITERATE
: DISP "SEED"
: Input x (input seed value)
: For (N,1,5,1) (loop for variable
: Disp 2x N starting at 1,
: 2x ->x ending at 5 with
: END increments of 1 unit)
Now run the program, using any seed. What is the fate of the orbit for each value?
[NOTE: This is a good time to explore the linking capabilities of the TI-82.]
Before continuing, do Class Activities numbers 1, 2 and 3.
Suppose you wish to borrow some money from a bank to purchase a car. The bank is charging 12% interest per year. Because of your financial situation, you can only afford a $100 car payment per month to repay the loan. What is the maximum amount of money you can borrow if you wish to pay off the loan in four years?
We will solve this problem using iteration. We will need two things, a function and a seed value.
Given: $100 = monthly payment
.01 = monthly interest rate (12% / 12)
Let: tn = time in n months
A(tn) = amount owed after n months
A(t0) = original amount of the loan (seed)
In words, the formula would look like the following:
Amount owed after 1 month = original amt. owed + interest due on amt. owed - payment
Amount owed after 2 months = amt. owed after 1 month + interest due on amt. owed - payment
Amount owed after 3 months = amt. owed after 2 months + interest due on amt. owed - payment over and over again.
In symbols, it would look like the following:
A(t1) = A(t0) + .01*A(t0) - 100
A(t2) = A(t1) + .01*A(t1) - 100
A(t3) = A(t2) + .01*A(t2) - 100
A(tn) = A(tn-1) + .01*A(tn-1) - 100
Factoring out A(tn-1) from the first two terms on the right side will simplify the expression:
A(tn) = (1 + .01)*A(tn-1) - 100
A(tn) = (1.01)*A(tn-1) - 100
This is the function we will iterate. All we need is a seed. Suppose we start with a $12,000 loan.
On the TI-82:
STEP 1: Go to Seq mode
STEP 2: Go to Y=
Un = 1.01 * Un-1 - 100
STEP 3: Go to TblSet
TblMin = 0
Tbl = 1
STEP 4: go to WINDOW
UnStart = 12000
nStart = 0
STEP 5: go to TABLE
Reading from the table, how much will you owe after the first month?...after the first year? What is the fate of the orbit of 12,000? What does that tell you about your loan?
Let's choose a different amount to borrow. Rework the problem for a loan of $10,000. To change the loan amount, you need to change the seed value. Go to WINDOW and let UnStart = 10000. Now display TABLE. What happens to the fate of this orbit? What happens to the loan balance?
Rework the problem for an $8000 loan. What is the balance after four years (48 months)? When will the loan be paid off? (You may wish to go to TBLSet and change Tbl to 12 as this will display only 12 month intervals)
Continuing this process, find the maximum amount you can borrow at 12% interest for 4 years making $100 monthly payments.
[Extensions: change the monthly payment, change the interest rate, or change the length of the loan to see what effect it has on the maximum amount one can borrow.]
The following problems were chosen for their value in using multiple representations (numeric, symbolic and graphic) to develop the idea of iteration. Some examples which lend themselves to the use of manipulatives are also included.
1. Your ancestors in the first, second and third generations are your natural parents, grandparents, and great-grandparents respectively. You have had 2 parents, 4 grandparents and 8 great-grandparents. How many ancestors do you have in the 10th generation back? Use iteration to determine your answer. Write your answer as a power. (Foerster; Algebra I, 1984)
2. Suppose a culture of 100 bacteria is put in a Petri dish and the culture triples every hour. Find when the number of bacteria will be 5,000,000.
3. Using either the ANS key method or the sequence mode method (Un = ...) iterate the function f(x) = x2. Use the following seed values and record your results.
a. x0 = 3
b. x0 = 1.1
c. x0 = 1
d. x0 = .8
e. x0 = 0
f. x0 = -.5
g. x0 = -1
h. x0 = -1.6
Determine the fate of each orbit and name any fixed points.
4. The Record Bar sells compact discs (C.D.'s) at the regular price of $14.95. The annual rate of inflation is 4%. If the inflation rate stays constant, how much will a C.D. cost after 3, 5, 7 and 9 years? If the inflation rate is 8%, what would the cost of a C.D. be after 10 years?
[Problem extension: Is it realistic to think that the price of a C.D. could be greater than $100?]
5. A ball is dropped from 30 feet above the ground. It bounces up to 75% of its previous height on each bounce. What is the height of the ball after 5 bounces? after 10 bounces? After how many bounces will the ball come to rest? Explain your reasoning.
6. Josh purchased a mountain bike for $350. He charged it on his Visa card which requires a minimum monthly payment of $15. Visa charges 1.5% interest on any unpaid balance. Using iteration, determine the number of months it will take to pay for the bike. How much did Josh end up paying for the bike and what was the total finance charge he paid?
7. Margarita has two options for investing her money. She can invest it at 5% annual compound interest for 10 years or she can invest for 5 years at 10% annual compound interest. Which investment will earn her more money? Which investment would be the wiser?
8. George places an initial deposit of $1000 in an annuity that earns 8% each year, and he deposits $500 at the end of each year. Write a recursive equation for the balance of the annuity at the end of each year. How much money will George have in the annuity after 10 years? 20 years? 30 years?
9. A person takes 400 mg. of ibuprofen every 4 hours. After 4 hours, the body eliminates 40% of the drug that is in the bloodstream. Write a recursive equation that describes this process. What is the long term level of ibuprofen in the bloodstream? (NCSSM; Calculus, 1993)
10. A butterfly population on an island declines by 20% each year. To keep the population from dying out, 50 butterflies are brought to the island and released at the beginning of each year.
a) Assuming that 500 butterflies are on the island initially, what is the long term trend in the population?
b) Suppose a severe winter reduces the butterfly population to 100. What will happen to the population in subsequent years?
c) What is the relationship of the initial population to the long-term trend in the population? (NCSSM; Calculus, 1993)
11. Suppose a rabbit population is 500 in January. Imagine two different rates of change for the population.
a) The population is increasing at the rate of 50 rabbits per month.
b) The population is increasing at the rate of 10% per month.
Compare the growth of the rabbit population over time given these two different growth rates. Which is faster? Use the sequence mode and sketch a graph of each on the same axis. (Let one be Un and the other Vn.) What is the difference in the graphs? (NCSSM; Calculus,1993)
12. The estimated cost of attending a state university for 4 years is $36,000. College costs are rising at the rate of 11% per year. Anjali is 5 years old. Her parents have three options for investing their money: a standard savings account, a Certificate of
Deposit, and US savings bonds. How much should they invest in each one of the three options in order to have enough for her education when she turns 18? (Hint: first determine what her college education will cost.)
a) A savings account earning 3.2% annual interest
b) Certificate of Deposit earning 4.5% annual interest
c) US savings bonds earning 5.7% annual interest.
13. Imagine taking a large piece of paper which is .003 inches thick. Cut the sheet into two sheets, placing one on top of the other. How tall is the stack? Repeat the process: Take the two pieces, cut each in half and stack all four on top of each other. How tall is the stack now? Continue this process for a total of 50 times. How tall is the stack? Estimate the height before doing the iteration. Compare this distance to the round trip distance to the moon. Is it possible to model this situation?
14. In the puzzle called The Tower of Hanoi, the goal is to move the tower of disks, one at a time from one peg to another peg without ever putting a larger disk on top of a smaller disk. There is a third peg to aid in moving. Determine the minimum number of moves needed to move a tower of 1, 2, 3, 4, and 5 disks. Using this information, create a table to show the number of disks and the minimum number of moves needed to move the disks. Use iteration to determine the minimum number of moves for 64 disks. (Hint: use the manipulative.)