|
|
Sequences on a TI-83 |
| © MathsNet 2001 |
| A series of activities on sequences using a TI-83 | |
| Written to complement
training materials for teachers on using graphical
calculators. |
  |
| 8. Some special sequences | |||||||||||||||||||||
A. The Fibonacci sequence This sequence 1, 1, 2, 3, 5, 8, 13... is known as the Fibonacci sequence. Each term is the sum of the two previous terms.
To set up your calculator to work with this recursive sequence, select MODE and on the fourth line select Seq. Now select Y= and you can enter the definition of the sequence. To enter u press 2nd 7. Note that you must enter the first two terms of the sequence in the line u(nMin):
Check by selecting TABLE (2nd GRAPH) that the sequence is correctly entered. Now look at the ratios of consecutive terms, ie., the values of 1÷1, 2÷1, 3÷2, 5÷3, 8÷5 and so on. Do this by returning to Y= and using the definition for v(n). To make your calculator do this correctly without creating errors, you must enter the equations exactly as shown below. The first two values of v(n) are made zero:
Scroll down the values of v(n). You should find that they tend towards the value 1.618033989. This number is known as the Golden Ratio. Its precise value is (1+Ö5)/2 and it is one root of the equation x2-x-1 = 0. In Section 4 you found out how to calculate differences. Find out the differences for the Fibonacci sequence:
What do these differences tell you about the formula for the nth term of the Fibonacci sequence? B. A chaotic sequence The recursive equation xnext = rx(1-x) is known to have some interesting properties, where r is a constant that you can change. Investigate the sequence defined by this recursive formula: u(1)=0.4, u(n) = Ru(n-1)(1-u(n-1)), and set R = 2.7 to start with. Select WINDOW and set nMax and Xmax to 200. Display a graph by selecting ZOOM... ZoomFit.
Now experiment with different values of R. In each case use WINDOW to choose the most appropriate scale for the y-axis. Concentrate on values of R ranging from 2.7 up to 4.
|
