- Answer the problems on the exam pages.
- There are five problems on pages 2-7, for 100 total points plus 10 extra credit. Actual scale was A=90, C = 54.
- If you need extra space use the back of a page.
- No books, notes, calculators, or collaboration.

Q1: 20 points Q2: 20 points Q3: 20 points Q4: 20 points Q5: 20 points plus 10 extra credit Total: 100 points plus 10 extra credit

Questions 1 and 2 deal with the **Fibonacci sequence** from Discussion
#5. Recall that the function F from naturals to naturals is defined
recursively, by the rules F(0) = 0, F(1) = 1, and (for n > 1) F(n) =
F(n-1) + F(n-2). As we calculated in the discussion, F(2) = 1, F(3) = 2,
F(4) = 3, and F(5) = 5.

**Question 1 (20):**Define SE(n) to be the sum of the*even-numbered*Fibonacci numbers from F(0) through F(2n). -- that is, SE(n) = ∑^{n}_{i=0}F(2i). (Thus, for example, SE(2) = F(0) + F(2) + F(4) = 0 + 1 + 3 = 4.) Prove, by ordinary induction on n, that SE(n) = F(2n+1) - 1.**Question 2 (20):**Prove by ordinary induction that for any natural n, F(6n) is congruent to 0 modulo 4 (that is, F(6n) ≡ 0 (mod 4)) and F(6n+1) is congruent to 1 modulo 4 (that is, F(6n+1) ≡ 1 (mod 4)). (Hint: Your induction should use both statements about n together to prove both statements about n+1.)**Question 3 (30+10):**In my pocket I have a large number of dollar bills and exactly six cents in change. Chocolate bars cost $1.43 each. I want to buy some number x of chocolate bars such that I can pay for them with my six cents of change and some of my dollar bills -- that is, I want x(1.43) to be exactly 0.06 more than some natural number.- (a,5) Formulate an equation in modular arithmetic that expresses my goal for the number x.
- (b,5) Carefully state the Inverse Theorem and explain how it applies to this situation.
- (c,10) Prove that the integers 43 and 100 are relatively prime.
- (d,10) Find both an inverse of 43, modulo 100, and an inverse of 100, modulo 43. Use one of these to find a natural number x that meets my goal stated above (and in part (a)).
- (e,10 XC) Suppoe how that the choclolate bars cost a cents and I have b cents worth of change in my pocket, where a is a natural and b is a natural less than 100. Exactly what conditions on a and b will guarantee that there exists an x such that I can buy x chocolate bars for some number of dollars plus b cents? I want an "if and only if" condition, so that if a and b meet your condition then x exists, and if they don't meet your condition then x doesn't exist. (Hint: If the process you used in (d) is not possible, how might you still be able to solve the problem?)

**Question 4 (30):**This problem deals with a directed graph G where each node N_{i,j}is given by a pair of naturals i and j. We can picture G with node N_{i,j}drawn at the position x=i and y=j in the usual Euclidean plane. There are four kinds of arcs (directed edges) in G:- For all even i and all j, there is an arc from N
_{i,j}north to N_{i,j+1}. - For all odd i and all j, there is an arc from N
_{i,j}south to N_{i,j-1}. - For all even j and all j, there is an arc from N
_{i,j}east to N_{i+1,j}. - For all odd j and all i, there is an arc from N
_{i,j}west to N_{i-1,j}.

(Note: This directed graph models the pattern of one-way streets in large parts of Manhattan Island.) It is not important to the problem how far G extends.

Here is an ASCII-art picture of part of G:

N03 <--- N13 <--- N23 <--- N33 <--- ... ^ | ^ | | | | | | v | v N02 ---> N12 ---> N22 ---> N32 ---> ... ^ | ^ | | | | | | v | v N01 <--- N11 <--- N21 <--- N31 <--- ... ^ | ^ | | | | | | v | v N00 ---> N10 ---> N20 ---> N30 ---> ...

Recall the inductive definition of paths in a directed graph:

- (1) For any node x, there is a path of length 0 from x to x.
- (2) If there is a path α of length k from x to y, and an arc e from y to z, then there is a path αe of length k+1 from x to z.
- (3) The only paths are those constructed from (1) and (2). Thus if a predicate P(α) is true for the empty path, and P(α) → P(αe) for any such path α and arc e, then P(α) is true for all α.

- (a,10) Prove by induction on all paths that if α is a path from
N
_{0,0}to N_{i,j}, then the length of α is greater than or equal to i+j. - (b,10) Explain (informally but carefully) why there is a path of length
exactly i+j from N
_{0,0}to N_{i,j}if i is even, if j is even, or if both are even. - (c,10) Prove by induction on all paths that if α is a path from
N
_{0,0}to N_{i,j}and i and j are*both odd*, then the length of α is greater than or equal to i+j+2. You may use the result of part (a).

Last modified 29 October 2007

- For all even i and all j, there is an arc from N