## 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.