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### Triangular Numbers

Date: 08/23/2003 at 20:44:13
From: Cough
Subject: Triangle numbers

How do I show that the sum of any two consecutive triangular numbers
is always a square number?

I can't seem to show triangular numbers as an equation.

1+2=3
1+2+3=6
1+2+3+4=10
1+2+3+4+5=15
1+2+3+4+5+6=21

1+3=4=2^2
3+6=9=3^2
6+10=16=4^2

Date: 08/23/2003 at 23:37:24
From: Doctor Stephanie
Subject: Re: Triangle numbers

Hi there!

One thing that might make this problem a lot easier for you is to
remember the formula for triangular numbers. If you want to find the
"nth" triangular number, the formula you would use would be:

N(N+1)
-------- = Nth triangular number
2

Now try to use the formula to find the "Nth" triangular number and the
"N+1st" triangular number. Then try adding the two consecutive
triangular numbers together and simplify, and I think you will find
what you are looking for.

If you need me to continue on the problem a little bit further, or
if you need more explanation, feel free to email me back!

- Doctor Stephanie, The Math Forum
http://mathforum.org/dr.math/
Associated Topics:
High School Number Theory

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