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**espeon****Real Member**- Registered: 2006-02-05
- Posts: 2,586

Umm.You know the link in 'introduction' about triangular numbers.Unfortunately I don't understand what it meant.Sorry!Can you help me.:/

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

You must tell what exacly you don't understand.

IPBLE: Increasing Performance By Lowering Expectations.

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**espeon****Real Member**- Registered: 2006-02-05
- Posts: 2,586

Well Ihave no idea what it was talking about.After 5 minuites of trying to read it all of the writing was like squiggly lines and I became dizzy.I tried again and the exact same thing happened.I can't work out out what the writing says.Please help me

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

Can you give me some link to this page.

I want to see if I could understand it.

IPBLE: Increasing Performance By Lowering Expectations.

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**espeon****Real Member**- Registered: 2006-02-05
- Posts: 2,586

I don't know how but go to introductions then to the topic made by fgarb.Ganesh put in a link to it

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

It seems clear to me. I can't only understand the "simplex" thingy, but I will.

I understand now why you can't unerstand this.

IPBLE: Increasing Performance By Lowering Expectations.

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**fgarb****Member**- Registered: 2006-03-03
- Posts: 89

Maybe it would help to start with something simple - do you understand what they are?

Try going back to the link left by Ganesh, and don't look at any of the text if it gives you a headache, just look at the table on the right. This table basically defines triangular numbers - they're numbers that can be displayed like the triangles shown there.

For example, the first triangular number is just 1 point, but that's a really boring triangle. The second triangular number uses two rows, but to make it look like a triangle, the second row needs one more point than the first row, so you have three total points - one from the first row, and two from the second row. So the second triangular number is three, and so on.

Does that makes sense? If you follow the pattern, you end up making a list of triangular numbers that looks like:

1,3,6,10,15,...

You get this list by starting with 1, then adding 2, then adding 3, then adding 4, ...

That's a starting point at least. If you understand this post then you know what triangular numbers are, and then we can work from there to help you understand their properties. I think they're pretty cool, so hopefully you'll enjoy learning about them.

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

She may understand the first part of the page but she won't understand the equations with binomials.

IPBLE: Increasing Performance By Lowering Expectations.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 18,077

A figurate number is a number that can be represented by a regular geometrical arrangement of equally spaced points. If the arrangement forms a regular polygon, the number is called a polygonal number. The polygonal numbers forming a triangle, square, pentagon, hexagon etc. are called triangular, square, pentagonal, and hexagonal numbers, respectively.

A triangular number is figurate number obtained by adding all positive integers less than or equal to a given positive integer n. The first few triangle numbers are 1, 3, 6, 10, 15, 21, ... etc.

Triangular numbers:-

* * * * *

* * * * * * * *

* * * * * * * * *

* * * * * * * *

* * * * *

1 3 6 10 15

Triangular numbers are of the form n(n+1)/2. The nth triangular number is n(n+1)/2. For example, the 10th triangular number is 10(11)/2 = 55.

Triangular numbers are also the sum of the first n natural numbers. For exmple, the fifth triangular number is 5(6)/2 = 15. It can be seen that the sum of the first 5 natural numbers is 1+2+3+4+5=15.

Some triangular numbers are also perfect numbers like 6, 28 etc. A perfect number is one whose sum of the factors including one, but excluding itself, is equal to the number.

Character is who you are when no one is looking.

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**espeon****Real Member**- Registered: 2006-02-05
- Posts: 2,586

I get the first bit of the link and I know the basics but I REALLY don't want to draw a triangle made of 100 rows.I want to find the 100th tri no. My dad said its 5050 and if I use ganesh's formula I get the same answer but somewhere it said n(n-1)/2 and I got 4950.I don't really know which one is right.Which one?

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

Will you trust to ganesh or to the other source?

Of course it's:

IPBLE: Increasing Performance By Lowering Expectations.

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**fgarb****Member**- Registered: 2006-03-03
- Posts: 89

Espeon, I have a story you might find interesting:

My junior high math teacher once told us about a famous mathematician in elementary school (I can't remember who the mathematician was, I'll call her Janet). Janet's teacher was getting tired of teaching, so one day she assigned the class a quiz in which they had to add up all the numbers from 1 to 500. Janet didn't know it then (she hadn't learned much math yet), but she was being assigned to calculate the 500th triangular number.

Janet, of course, didn't know any triangle number formulas, and she thought adding up all those numbers would be really boring. So, while the other students started working, she instead did a bit of doodling. Without really thinking about it, she wrote out the problem twice on her page.

001 + 002 + 003 + ... + 500 = ?

001 + 002 + 003 + ... + 500 = ?

Great, if I add all those numbers up, I'll do twice as much work and get twice the answer that I'm looking for, she thought. Bored, she erased the second line and rewrote it backwards, like this:

001 + 002 + 003 + ... + 500 = ?

500 + 499 + 498 + ... + 001 = ?

That's strange, Janet thought, if I add those numbers vertically, they all add up to 501 (did you notice?). 1+500=501, 2+499=501, 3+498=501, etc. But, she also knew that if she added up all of the numbers, all 1000 of them, she'd get double the answer her teacher wanted. So she wrote a third line:

001 + 002 + 003 + ... + 500 = ?

500 + 499 + 498 + ... + 001 = ?

------------------------------

501 + 501 + 501 + ... + 501 = double ?

So if you add 501 to itself 500 times, you'll get twice the answer that you're looking for, Janet realized. But adding something to itself lots of time is just the same as multiplication. This meant that 500 multiplied by 501 was double the answer she wanted:

500 x 501 = 2 x ?

Suddenly, Janet knew how to get the answer. She realized this meant ? had to be half of 500 x 501. So Janet solved the problem:

[align=center]

[/align]She wrote down the answer and turned her quiz in. It took the other students more than an hour to finish, but Janet was done in ten minutes, and she was the only person who got the right answer - everyone else made a mistake adding all those numbers together.

You might notice that this technique could work for any triangular number, not just 500. Let's say you're looking for the Nth triangle number. Then,

[align=center]

[/align]This works for all triangle numbers. If you plug in N = 500, you'll get Janet's formula!

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

Good explanation fgarb.

IPBLE: Increasing Performance By Lowering Expectations.

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

I think the person in fgarb's story was Gauss.

Why did the vector cross the road?

It wanted to be normal.

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

i too.

IPBLE: Increasing Performance By Lowering Expectations.

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

Cool story, fgarb!!! I never heard of that double backwards method; it is beautiful!!

**igloo** **myrtilles** **fourmis**

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

I also thought of this slightly different way too, fgarb!!

You include zero in the counting numbers, so

0 + 1 + 2 + 3 ... + 500

500+499 +498+497...+0

___________________________+

Add that all up and you still get 500 times 501 for double the total.

Pretty neat huh?

Here is it without zeros:

1 2 ... 500

500 499 ... 1

__________________+ and you get 501 times 500, the same thing!! I love it.

**igloo** **myrtilles** **fourmis**

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

It may be done fully geometrically:

* ****

** ***

*** **

**** *

The horisontal is x+1, the vertical is x, the sum is half.

IPBLE: Increasing Performance By Lowering Expectations.

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