Math 1311 Term Paper Final Spring 2018 Prof. SIBNER
1. The Fibonacci sequence uses the recurrence relation Fn=Fn-1+Fn-2 with starting seeds 1 and 1. The limit ratio is (1+5)/2. A variation is to use the seeds 1 and 2, resulting in the Lucas sequence.
A. Find the limit ratio if one uses the seeds 3 and 17.
B. Other than varying the seeds, describe at least one other variation that we discussed in class. For one of the examples that you give, find the equation for the limit ratio. (Do not solve the equation.)
C. Explain why our example of plant propagation results in a “Fibonacci generalization” recurrence relation.
2. In our discussion of various topics, we encountered some theorems and some conjectures.
A. Describe at least two conjectures that arose in the discussions of each of two different topics; one of the two topics should involve numbers.
B. For the topic involving numbers, give several examples in support of each of the two conjectures.
C. State a theorem for each of these two topics.
3. Explain how using a rectangle with sides identified is an abstract representation of various surfaces. Draw rectangles with various sides identified, representing (a) sphere, (b) cylinder, (c) torus, (d) Mobius band, (e) Klein bottle.
4. Give two examples of how we used an “argument by contradiction” during the course. Give the arguments.
5. Explain what the Euler characteristic has to do with Platonic solids.
6A. Describe the various possible symmetries of a strip. (A strip always has translation symmetry.)
B. Which of the symmetries can appear on a Mobius band?
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