 
Summary: Math 3130 Midterm: Due Friday October 28 in Class
Remember to show your work!
Directions This is a take home, open notes test. There are eight problems on the test. You must
answer carefully and completely any five of the problems. You answer must be your own; you
are free to discuss problems with other students but must write up your own answers. Excessive
similarity of answers will be noted and dealt with appropriately.
Problem 1 Prove by induction that
n
k=1
k3
=
n2
(n + 1)2
4
Problem 2 Find all possible cycle types in S5 and say how many permutations there are with
each of these cycle types.
Problem 3 Compute GCD(91, 56) = 91x + 56y using the Euclidean algorithm and solving for
x and y by back substitution through the Euclidean algorithm.
Problem 4 Let A be a group with two elements and so that A consists of all possible products
in any order of and . If and satisfy the following relations:
