Mathematics from high school to university
Creative problem solving, with fun tricks and geometrical illustrations of the results.
How to derive the formula for the sum of squares of all natural numbers from 1 to n
How to derive the formulas for partial sums for two very important examples of arithmetic progressions: 1+2+…+n and 1+3+5+…+(2n-1).
How to derive the general formula for partial sums of geometric progressions; an important example of the sum of all the natural powers (from 1 to n) of 1/2.
How to derive some less known formulas, using cool computational tricks.
How to prove divisibility in some simple cases, using very basic (but fun) reasoning.
You will learn (from an article) how to convert infinite periodic decimal expansions into fractions, by smart substitutions and solving linear equations.
You will learn (from an article) some basics about continued fractions (CF), and how to recognise irrational surds from their infinite periodic CF-expansions.
You will learn (from an article) about an interesting family of nested square roots and how to prove that some members of this family are… integers!