More problems

The second exam (Tuesday before Thanksgiving) will be over generating functions, counting problems, and complex numbers. Here are some questions to work on in preparation.

1. The generating function for a seqence [Maple Math] is [Maple Math] . Find the sequence.

2. Suppose [Maple Math] is the sequence defined recursively by [Maple Math] for n > 0, [Maple Math] . Find a closed formula for [Maple Math] .

3. Suppose [Maple Math] is the sequence defined recursively by [Maple Math] , [Maple Math] , and for n > 1, [Maple Math] . Find a closed formula for [Maple Math] .

4. Prove that [Maple Math]

5. Find the coefficient of [Maple Math] in [Maple Math] .

6. Suppose food baskets with 12 items of three kinds (canned goods, root crops, and meats) are to be made up for Thanksgiving. Each basket must have an odd number of cans (green beans, cranberry sauce, etc), an even number of potatoes, turnips etc, and an odd number of instances of meat (but no more than 5 pieces). If no family is to receive the same type of basket (that is, no two baskets have the same number of each kind of food), how many families could get a basket this halloween?

7. Find the number of positive solutions to [Maple Math] .

8. Compute and express in the form [Maple Math] or [Maple Math] . Draw them on a coordinate graph.

a) [Maple Math] b) [Maple Math] c) The sum of all the Gaussian integers [Maple Math] , with i and j between 0 and n.

d) the square roots of 1+ I e) the 5th roots of 1. f) the sum of the positive powers of [Maple Math] .

9. a) Solve [Maple Math] for z. b) Find the imaginary part of the conjugate of [Maple Math] , where [Maple Math] .

c) Solve [Maple Math] for z by completing the square. Express the roots in the form [Maple Math]

10. Show that if p(z) is a polynomial with real coefficients then the non-real roots of p(z) occur in conjugate pairs: that is, if [Maple Math] then [Maple Math] . Prove the converse, if its true.

11. Suppose p(z) is a polynomial such that if [Maple Math] then [Maple Math] . Show that the polynomial [Maple Math] has real coefficients.

12. Must a quadratic polynomial with real fixed points have real coefficients?

13. Show every cubic polynomial has at least one real fixed point.

14. Find the area of the image of the circle of radius 1 centered at 2+I under inversion [Maple Math] .

15. Find the fixed points of [Maple Math] considered as a Moebius map on the extended complex plane.

16. Show that every Moebius map has at least one fixed point (when considered as a map on the extended plane).

17 Which Moebius maps have only 1 fixed point?