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In this course, you will learn:
- Problem solving that is creative, with interesting tricks and geometrical pictures of the results.
- How to calculate the sum of squares of all natural integers from 1 to n.
- How to develop partial sum formulas for two key arithmetic progressions: 1+2+...+n and 1+3+5+...+(2n-1).
- The universal formula for partial sums of geometric progressions; an essential example of the sum of all natural powers (from 1 to n) of 1/2.
- How to use interesting computational strategies to generate some lesser-known formulas.
- How to show divisibility in certain simple circumstances using simple (but entertaining) logic.
- Learn how to transform infinite periodic decimal expansions into fractions using smart replacements and linear equations (from an article).
- The fundamentals of continuous fractions (CF) and how to identify irrational surds from their infinite periodic CF-expansions (from an article).
- About an interesting family of nested square roots (from an article) and how to prove that some members of this family are... integers!
Syllabus:
- Derivations of some formulas
- Some proofs without induction
- Three groups of problems where we avoid infinite repetitions in a smart way
