91Ó°ÊÓ

Each of the numbers $$ 1=1,3=1+2,6=1+2+3,10=1+2+3+4, \ldots $$ represents the number of dots that can be arranged evenly in an equilateral triangle: $$ \therefore \cdots $$ 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 \(n \geq 1\). (Pythagoras, circa 550 B.C.) (b) The integer \(n\) is a triangular number if and only if \(8 n+1\) is a perfect square. (Plutarch, circa 100 A.D. (c) The sum of any two consecutive triangular numbers is a perfect square. (Nicomachus, circa \(100 \mathrm{AD}\).) (d) If \(n\) is a triangular number, then so are \(9 n+1,25 n+3\), and \(49 n+6 .\) (Euler, 1775 )

Derive the following formula for the sum of triangular numbers, attributed to the Hindu mathematician Aryabhatta (circa 500 A.D.): $$ t_{1}+t_{2}+t_{3}+\cdots+t_{n}=\frac{n(n+1)(n+2)}{6} \quad n \geq 1 $$ [Hint: Group the terms on the left-hand side in pairs, noting the identity \(\left.t_{k-1}+t_{k}=k^{2} .\right]\)

In the sequence of triangular numbers, find the following: (a) Two triangular numbers whose sum and difference are also triangular numbers. (b) Three successive triangular numbers whose product is a perfect square. (c) Three successive triangular numbers whose sum is a perfect square.

(a) If the triangular number \(t_{n}\) is a perfect square, prove that \(t_{4 n(n+1)}\) is also a square. (b) Use part (a) to find three examples of squares that are also triangular numbers.