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.
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?
BonusTry 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.