Each of the numbers 1 = 1,3 = 1+2.6= +2 + 3, 10 = 1 + 2 + 3 + 4,? represents the

Question

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.)

Explanation / Answer

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

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