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:

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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?

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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?

1. 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.