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Triangular Numbers (Part I)

Going back in history more than two thousand years. Pythagoras and the Pythagoreans were interested in establishing relations between geometric figures and numbers.

If you take one, one and two, one two and three stones and arrange them in a particular way. You may be able to produce triangular figures. This is now believe to be the origin of the triangular numbers“, “the square numbers and other figured numbers.

Triangular numbers are obtained when we arrange a number of stones in an equilateral triangular shape. As we can see in the figure we have the first three triangular numbers. One, three and six. The first triangular number is one. The second triangular number is three. To obtained three we need to add one and two. To get the third triangular number. We add one plus two plus three.

Can you guess now the next triangular number?

Yes, you guess right!

The next triangular number is ten. Ten is the result of adding one plus two plus three plus four. So there is an intimated relationship between triangular numbers and the sum of consecutive natural numbers.

If we call T_n the n^{th} triangular number, then we can define the n^{th} triangular number to be the number obtained when we add the natural numbers 1+2+3+ \cdots+n. From our definition of triangular numbers, we can see a direct link between triangular numbers and the sum of the first n consecutive natural numbers.

It will be nice to find a short-cut that will allow us to compute easily the sum of the natural numbers 1+2+3+ \cdots +n. A problem similar to this was solve a long time ago by a very young child. He became later in life one of the greatest mathematicians in history.His name was Carl Friedrich Gauss.

The story goes that his teacher Butner, wanted the students busy working on additions for a while so he asked the students to compute a lengthy sum and for that he choose the numbers in a way that he could easily verify the answers himself, without having to do the full computation.

To his astonishment one of the students finished the addition with the correct answer in a very short time. Obviously that student could not have possibly added the one hundred numbers in such a short time. That student had to have a very simple and brilliant idea that allowed him to easily solve the problem.

The teacher immediately understood how bright this student was and from that moment forward pay special attention to develop that student’s talent.

If you have not seen this problem before. Now you have the opportunity to solve the same problem that Gauss solved when he was young! Stop the lecture and try finding the sum of the first one hundred natural numbers. That is,

Find the sum

1+2+3+\cdots+100

(This is a partial transcription of the Video Lecture Triangular Numbers (I) the video continues displaying Gauss’s Solution)

The complete video lecture can be seen at Triangular Numbers (I)

This is a blog posting from www.isallaboutmath.com

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