Homework 4
Fibonacci Numbers & Related Topics
Due Thursday, February 15, 2007
#1. Prove that
n
k=0
n
Fk = F2n .
k
#2. Consider the innite series
2
3
4
5
6
7
Fn xn .
0 + 1x + 1x + 2x + 3x + 5x + 8x + 13x + =
i=0
Find the radius of convergence of this serie
Homework 2
Fibonacci Numbers & Related Topics
Due Thursday, February 1, 2007
Note: On problems 1, 2, and 3, you must explain your reasoning. It is not enough just to
write down a numerical answer.
#1. Let n be a xed positive integer. Find the probability
Homework 5
Fibonacci Numbers & Related Topics
Due Thursday, March 22, 2007
#1. Using generating functions, solve the dierence equation
an+2 an+1 2an = 0,
with initial conditions a0 = 9 and a1 = 3.
#2. Using generating functions, solve the dierence equatio
Homework 1
Fibonacci Numbers & Related Topics
Due Thursday, January 25, 2007
#1. Calculate the rst 10 terms of the sequence given by T0 = 0 and Tn = Tn1 + n for
all n 1. The numbers in this sequence are called triangular numbers. If we have time,
well see
Homework 3
Fibonacci Numbers & Related Topics
Due Thursday, February 8, 2007
#1. Prove that for all positive integers n, we have F2n = Fn Ln .
Hint: Weve done almost a complete proof of this in class (although I didnt tell you that
at the time). To prove
Homework 6
Fibonacci Numbers & Related Topics
Due Thursday, March 29, 2007
#1. Consider the proof that we gave in class of Theorem 22. Explain why this proof does
not work if m = 2. Do not give me counterexamples to the theorem when m = 2. Go carefully th
Homework 8
Fibonacci Numbers & Related Topics
Due Thursday, April 12, 2007
#1. Prove that 17 does not divide any odd Fibonacci numbers.
#2. Prove that if n > 4 and n is composite then Fn is also composite.
Hint: If n > 4 and n is composite, then n has at
Homework 10
Fibonacci Numbers & Related Topics
Due Thursday, April 26, 2007
#1. Write each number from n = 30 to n = 40 as a sum of Fibonacci numbers.
#2. Write each number from n = 85 to n = 95 as a sum of Fibonacci numbers without
using the number F9 =
Homework 9
Fibonacci Numbers & Related Topics
Due Thursday, April 19, 2007
#1. Prove that if m, n 1, then Lm+n = Fn+1 Lm + Fn Lm1 .
#2. Prove part b) of Corollary 34 by using the Binet Formulas.
#3. Find all of the Lucas numbers which are divisible by 29.
Homework 7
Fibonacci Numbers & Related Topics
Due Thursday, April 5, 2007
#1. Let m be an integer greater than 1. Prove that congruence modulo m is an equivalence
relation.
#2. Suppose that a, b, c, d Z, that a c (mod m), and that b d (mod m). Prove that: