Each of the numbers 1 13 126 2 3 10 1 2 3 4 represent

Each of the numbers 1 = 1,3 = 1+2.6= +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 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.)

Solution

a) The nth number T(n) = 1 + 2 + 3 + ... + n.

then

T(n)+T(n)

T(n)+T(n) = n(n+1)

2T(n) = n(n+1)

T(n) = (1/2)n(n+1)

b)

n = 1+2 +3 +.... T(n) = (1/2)n(n+1)

Now multiply n by 8 and add 1
8n+1 =8[(1/2)x(x+1)] +1
4x(x+1) +1
4x² +4x +1
8n+1 = (2x +1)²

So, 8n+1 is a perfact square.

c)

Let first consecutive triangle number = n(n+1)/2

then second consecutive triangle number = (n+1)(n+2)/2

Sum of theses two number = n(n+1)/2 + (n+1)(n+2)/2

= (n² + n + n² + 3n + 2) / 2

= (2n² + 4n + 2)/2

= n² + 2n + 1

n(n+1)/2 + (n+1)(n+2)/2 = (n+1)²

T(n)+T(n)	=	1 +	2 +	3 +	... +	(n-1) +	n
	+	n +	(n-1) +	(n-2) +	... +	2 +	1