Posts Tagged ‘Fibonnacci’

Computing a correct answer is sufficient for most grade school work, but proving that something must be true is one of the most important skills our students can learn.  This ability has immediate benefits for such important tests as USAMTS or the USAMO as well as teaching the skills of reasoning and communication that can help in any profession.

We are very fortunate to once again have a lecture by Dr. Branislav Kisačanin.  Here is his description of the upcoming session:

Introduction to Mathematical Proofs
In this lecture we will see examples of common proof techniques: direct proofs, proofs by induction, and proofs by contradiction. The students should have their pencils ready, as they will get a hands-on experience solving and proving mathematical problems.
In addition to a few more problems of historical significance, we will work on relatively simple problems that illustrate various proof techniques. In order to make things more interesting, we will let the students figure out which technique to use for each problem.

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The 2010-2011 season of Metroplex Math Circle will have a strong start with our first lecture by Dr. Branislav Kisačanin.

Dr. Kisačanin is a computer scientist at Texas Instruments working in the field of computer vision. In his spare time he likes to use computer vision for fun project such as his Tetris-playing robot. He loved math and physics competitions and nowadays likes to challenge his wife, kids, and friends with math problems and puzzles. Dr.Kisačanin wrote a book about selected mathematical gems:Mathematical Problems and Proofs.

Dr. Kisačanin has chosen Number Sequences as his first topic.  These sequences are critical parts of the tool kit of any middle school problem solver but also offer unexplored mysteries for professional mathematicians.  Following are some of the topics Dr. Kisačanin may cover in the course of his talk:

Triangular numbers

  • Derive the formula and the relation to square numbers both algebraically and geometrically
  • Sum of n squares (Gregory’s vanishing triangle, induction)
  • Sum of n cubes (connection back to triangular numbers)
  • Excercise: Sum of n odd numbers (again alg + geom)
  • OPTIONAL: Sums of higher powers (telescoping)

Fibonacci numbers

Binomial coefficients

  • Coefficients in (a+b)^n
  • Combinations
  • Pascal’s Triangle
  • Excercise: each row sums to 2^n, finding triangular and Fibonacci numbers, …

Other sequences and open problems

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