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University of Manitoba, Mathletics 2009
Seesion 1: Mathematical induction, 15 September 2009
1
1.1
Facts and denitions
The two main principles of mathematical induction
Mathematical induction (abbreviated MI) is a proof technique that applies to many math
2009 University of Manitoba Mathletics Training
WEEK 6: Sequences and Recursion (Draft: October 21, 2009)
A sequence is a nite or innite list of numbers, often denoted formally with notations like
cfw_an ; cfw_an N ; a1 , a2 , a3 , . . . , an , . . . ; o
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University of Manitoba Mathletics
All Day Practise #1 Nov. 5, 2005, 912
INSTRUCTIONS
These problems are designed to be fun as well as challenging. Partial credit will be given for
signicant progress, but a thorough job on a few problems is worth more
Linear Algebra II
Mathematics 136.235
Assignment #1 Key
November 12, 2008
2.1: 7 Prove (A + B)C = AC + BC for (real) matrices A, B, C (whenever these operations are
dened).
Proof:
(A + B)C =([aij ] + [bij ])[cij ]
(index notation)
=[aij + bij ][cij]
=
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Linear Algebra II
Mathematics 2352
Assignment #2 Key
December 22, 2008
NOTES: Im making a key for this one because Im still not seeing what I expect on paper.
Mostly, Im seeing far too much. Some of you appear to feel that lots of detail make a good solut
Number Theory
University of Manitoba, Mathletics 2009
1
Facts and denitions
Modular Arithmetic: Let a, b, and m be integers, with m = 0. We say that a and b are
congruent modulo m if m divides a b. We denote this by a b (mod m).
Properties:
a a (mod m) (