## 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?

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.

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

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

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

ReplyDelete