|
|
| Line 1: |
Line 1: |
| − | Hi, I'm Colball. | + | = Hi, I'm Colball. = |
| | | | |
| − | You might know me as the founder of AoPS times or the founder of ''The Cool. The Amazing. The Poll Forum''. Anyway, I am going to talk about one of my favorite theorems. It says that <math>1+2+3+...+n+1+2+3...+(n-1)=n^2</math>. And here are three proofs: | + | You might know me as the founder of AoPS times or the founder of ''The Cool. The Amazing. The Poll Forum''. |
| | | | |
| − | PROOF 1: <math>1+2+3+...+n+1+2+3...+(n-1)=n^2</math>, Hence <math>\frac{n(n+1)}{2}+\frac{n(n+1)}{2}=n^2</math>. If you dont get that go to words.Conbine the fractions you get <math>\frac{n(n+1)+n(n-1)}{2}</math>. Then Multiply: <math>\frac{n^2+n+n^2-n}{2}</math>. Finnaly the <math>n</math>'s in the numorator cancel leaving us with <math>\frac{n^2+n^2}{2}=n^2</math>. I think you can finish the proof from there.
| |
| | | | |
| − | | + | == Now I will talk about myself: == |
| − | PROOF 2: The <math>1+2+\cdots+n</math> part refers to an <math>n</math> by <math>n</math> square cut by its diagonal and includes all the squares on the diagonal. The <math>1+2+\cdots+ n-1</math> part refers to an <math>n</math> by <math>n</math> square cut by its diagonal but doesn't include the squares on the diagonal. Putting these together gives us a <math>n</math> by <math>n</math> square.
| |
| − | | |
| − | | |
| − | PROOF 3: We proceed using induction. If <math>n = 1</math>, then we have <math>1+0=1^2</math>. Now assume that <math>n</math> works. We prove that <math>n+1</math> works. We add a <math>2n+1</math> on both sides, such that the left side becomes <math>1+2+\cdots + (n+1)+1+2+\cdots + n = n^2 + 2n + 1 = (n+1)^2</math> and we are done with the third proof.
| |
| − | | |
| − | Now I will talk about myself: | |
| | Age: <math>8<x<17</math> | | Age: <math>8<x<17</math> |
| | Joined AoPS: Check my Profile (it's my blog) | | Joined AoPS: Check my Profile (it's my blog) |
Revision as of 15:26, 16 June 2019
Hi, I'm Colball.
You might know me as the founder of AoPS times or the founder of The Cool. The Amazing. The Poll Forum.
Now I will talk about myself:
Age: 
Joined AoPS: Check my Profile (it's my blog)
FTW rating: Bad
Overall alumus level: 8ish
Gender: Can't tell you, (sorry)
Goals: Make longest AoPS wiki user page.
Goals Report: Almost.
Location: Tucson, AZ
Brothers username: kix
Post count: 58, (it would br more if my mom )
Joined AoPS: Check my Profile (it's my blog)
FTW rating: Bad
Overall alumus level: 8ish
Gender: Can't tell you, (sorry)
Goals: Make longest AoPS wiki user page.
Goals Report: Almost.
Location: Tucson, AZ
Brothers username: kix
Post count: 58, (it would br more if my mom )
