- #1
vanmaiden
- 102
- 1
Homework Statement
I'm not very proficient with LAtex, so I'll try to translate this mess the best I can. It's a summation
[itex]\sum[/itex] i2 (on the bottom, there would be an "i = 1") (on the top, there would be an "n") this summation equals [itex]\frac{n(n+1)(2n+1)}{6}[/itex]
In the summation, basically, "i" starts off at one and goes to "n"
Why does this summation equal [itex]\frac{n(n+1)(2n+1)}{6}[/itex]?
Homework Equations
The definition of a definite integral? I found this summation through doing definite integrals using Riemann sums. I found the answer to the summation online, but I wanted to know how one arrives at the answer.
The Attempt at a Solution
So far, I have done this:
S = 12 + 22 + 32 + ... + (n - 1)2 + n2
S = n2 + (n - 1)2 + (n - 2)2 + ... + 22 + 12
I just wrote out a part of the summation, and under it, I did the same summation part but in reverse.
Then...I added the two summations to get
2S = n2 + 1 + (n - 1)2 + 4 + (n - 2)2 + 9 + ... + (n - 1)2 + 4 + n2 + 1
I saw somewhat of a pattern, but it was hard to explain and I had to stop here. If someone could give me a link to an explanation, or if someone could fit an explanation in a response or two as to why this summation equals [itex]\frac{n(n+1)(2n+1)}{6}[/itex], that would be fantastic. Thank you in advance!