M328K Number Theory Unique No. 56115 Spring 2011
Dr. Bart Goddard Office: RLM 13.140 e-mail: goddardb@newsguy.com webpage: www.ma.utexas.edu/users/goddardb/home Hours: MWF 2:002:50 or by appointment Rules: Don't miss class. Don't be late. Don't buy plane
M328K Final Exam Solutions, May 10, 2003
1. `Bibonacci' numbers. The Bibonacci numbers
for n>2,
are defined by
,
, and,
.
a) Prove that, for all positive integers n,
.
The proof is by generalized induction. It is true for n=1 and n=2. Now suppose it is tr
M328K Final Exam, May 10, 2003
1. `Bibonacci' numbers. The Bibonacci numbers
for n>2,
are defined by
,
, and,
.
a) Prove that, for all positive integers n,
.
b) Prove that, for all positive integers n,
.
2. The following theorem-proof combination is erron
M328K First Midterm Solutions, February 21,
2003
1. Using induction, prove the formula:
We prove this by induction. The formula is true for the base case n=1, since
Now for the inductive step. We assume the formula is true
for n=m-1 and show it works for
M328K First Midterm Exam, February 21, 2003
1. Using induction, prove the formula:
2. As you know, the Fibonacci numbers
are defined by
,
and, for n>2,
. Give a rigorous proof of the assertion: ` is divisible by 3 if and only if n
is divisible by 4.' [Hin