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Triangular Numbers
By Anonymous on Saturday, February 19, 2000 - 01:10 am:

What I need to know (explained simply!) is what exactly triangular numbers are. Thank you.

By John Grindall (Jhg25) on Monday, February 21, 2000 - 07:01 pm:


Since you will know what square numbers are -
eg. 16 is a square number because 4 x 4 =16 we can draw this-


* * * *
* * * *
* * * *
* * * *


so 15 is a triangular number because -

*

* *

* * *

* * * *

* * * * *


like the red balls in a snooker triangle.

If you know the formula for the sum of the numbers
(1 + 2 + 3 + ... + n) then you can work out the n'th triangular number.
15= 1+2+3+4+5
21=1+2+3+4+5+6
etc.
By Anonymous on Monday, November 22, 1999 - 09:53 pm:

Is there a pattern or rule for these type of numbers??

By David Hodge (P1262) on Tuesday, November 23, 1999 - 09:20 am:

Yes. The nth triangular number is given by the formula (1/2)×n×(n+1).

By Dan Goodman (Dfmg2) on Tuesday, November 23, 1999 - 07:08 pm:

There is a simple proof of this. Say you want to find the nth triangular number, this involves adding up the first n numbers (because the first level of the triangle has 1 point, the second level has 2 points, the third 3, and so forth). Imagine writing out the first n numbers in a row, and underneath them, writing out the same list backwards, and underneath that, writing out the sum of the two numbers above it, e.g. for n=5 we would have:

1 2 3 4 5 (added up gives you 15)
5 4 3 2 1 (added up gives you 15)
---------
6 6 6 6 6 (added up gives you 30 = 6×5)

The bottom row always has the same number (n+1). Now, adding up both of the first two rows will give you twice the sum of the first n numbers. Adding up the first two rows is the same as adding up the bottom row. But this is easy to do, as all the numbers are the same (n+1) and there are n of them, so the sum of the bottom row is n(n+1). This is twice the sum of the top row, so 1+2+...+n=(1/2)n(n+1). Tada!

There's an interesting story that goes with this too. Apparently Gauss (one of the greatest mathematicians ever, arguably the greatest) was given the problem of adding up the first 100 numbers by his maths teacher at age 7 (the maths teacher was trying to keep him busy while he did something else). Gauss worked out the general rule in two minutes, and promptly gave the answer 5050 to his teacher's amazement.

By Anonymous on Wednesday, November 24, 1999 - 11:48 pm:

There is a recent series of problems on the NRICH website on triangular numbers. See
Cat Food and Man Food. They turn up over and over again as you will find if you do a search on the rest of the site. Ed