Tuesday, May 5, 2015

Triangle Number Worksheet

Triangle Numbers

What is the sum of 1 + 2 + 3?  

What is the sum of 1 + 2 + 3 ....   + 10?

What is the sum of  1+ 2 + ....  + 100?

What is the sum of 1 + 2 +  .... + 568?

We call such sums triangle numbers because they can always be drawn as follows:




Can you find a general formula for 1 + 2 ...  + N based on what you did above?  After that 
see if you can come up with a way to prove this is true. Hint: there are multiple ways to do so. See if you can a find a geometric explanation that takes advantage of the fact that triangle numbers are actually triangles.




Raise your hand once you've thought about or solved the above question.

Problems

1. Find an expression for the first n even integers: 2 + 4 + 6 + 8 + . . . + 2n

2. If there is a room full of 50 people and everyone shakes hands with every other person how many handshakes are there?  Can you generalize to if there is a room full of N people?

3. Amy has a box containing ordinary domino pieces (up to six dots) but she does not think it is a complete set. She has 24 dominoes in her box and there are 125 spots on them altogether.
Which of her domino pieces are missing?


4. One line can divide a circle into at most 2 regions, Two lines can divide a circle at most into 4 regions. Three lines can divide a circle into at most seven regions. What is the maximum number of regions that a circle can be divided into by 50 lines?

5. Try multiplying a triangle number times 8 and adding 1. What kind of number do you seem to find?  Can you figure out why this is happening?


Bonus

Try making a chart of the square of the triangle numbers and see if you can find a formula for it.
For example: (1 + 2 + 3) squared is 36.  Hint: take a look at the list of squares and cubes.

2 comments:

  1. If I reuse: this was interesting https://mathbabe.files.wordpress.com/2015/07/300px-triangularnumbers.gif?w=595

    In a Ulam spiral the triangle numbers form a 17 armed spiral.

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  2. Also worth adding fro every triangle number t 8t + 1 = K where K is a perfect square. This has a neat geometric representation.

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