Proofs without Words Dana C. Ernst Northern Arizona University Mathematics & Statistics Department http://danaernst.com Friday Afternoon Mathematics Undergraduate Seminar September 20, 2013 D.C. Ernst Proofs without Words 1 / 17
Theorem For all n ∈ N, 1 + 2 + · · · + n = n(n + 1) 2 . Note The numbers Tn := 1 + 2 + · · · + n are called triangular numbers. D.C. Ernst Proofs without Words 3 / 17
Theorem For all n ∈ N, 1 + 2 + · · · + n = C(n + 1, 2) := (n + 1)! 2!(n − 1)! . Corollary For all n ∈ N, C(n + 1, 2) = n(n + 1) 2 . D.C. Ernst Proofs without Words 4 / 17
Theorem For all n ∈ N, 1 + 3 + 5 + · · · + (2n − 1) = n2. Note This the same as the previous theorem, but with a different visual proof. D.C. Ernst Proofs without Words 6 / 17
Theorem Let Pn be the nth pentagonal number. Then Pn = 3Tn−1 + n. Note The nth pentagonal number is given by Pn := 3n2 − n 2 . D.C. Ernst Proofs without Words 7 / 17
Theorem The alternating sum of the first n odd natural numbers is n. In other words, for all n ∈ N, n k=1 (−1)n−k(2k − 1) = n. D.C. Ernst Proofs without Words 9 / 17
Theorem (Pythagorean Theorem) If a, b, c ∈ N are the lengths of the sides of a right triangle, where c the length of the hypotenuse, then a2 + b2 = c2. D.C. Ernst Proofs without Words 10 / 17
Theorem (Pythagorean Theorem) If a, b, c ∈ N are the lengths of the sides of a right triangle, where c the length of the hypotenuse, then a2 + b2 = c2. D.C. Ernst Proofs without Words 11 / 17
Sources All the figures in this talk came from the following locations: • MathOverflow: http://mathoverflow.net/questions/8846/ proofs-without-words • Art of Problem Solving: http://www.artofproblemsolving.com/Wiki/index. php/Proofs_without_words • Strogatz, NY Times: http://opinionator.blogs.nytimes.com/2010/04/04/ take-it-to-the-limit/ D.C. Ernst Proofs without Words 17 / 17