# Each of the numbers1=1, 3 = 1+2, 6 = 1+2 + 3, 10 = 1+2 + 3 + 4, ...represents the number

Chapter 2, Problem 1

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Each of the numbers1=1, 3 = 1+2, 6 = 1+2 + 3, 10 = 1+2 + 3 + 4, ...represents the number of dots that can be arranged evenly in an equilateral triangle: This led the ancient Greeks to call a number triangular if it is the sum of consecutiveintegers, beginning with 1. Prove the following facts concerning triangular numbers:(a) A number is triangular if and only if it is of the form n(n + 1)/2 for some n 2: 1.(Pythagoras, circa 550 B.C.)(b) The integer n is a triangular number if and only if 8n + 1 is a perfect square. (Plutarch,circa 100 AD.)( c) The sum of any two consecutive triangular numbers is a perfect square. (Nicomachus,circa 100 AD.)(d) If n is a triangular number, then so are 9n + 1, 25n + 3, and 49n + 6. (Euler, 1775)

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