Each of the numbersrepresents the number of dots that can be arranged evenly in an equilat...

Each of the numbers

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 consecutive integers, 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 > 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 A.D.)

(c) The sum of any two consecutive triangular numbers is a perfect square. (Nicomachus, circa 100 A.D.)

(d) If n is a triangular number, then so are 9n + 1, 25n + 3, and 49n + 6. (Euler, 1775)