Question

How do you derive this?\[ \sum_{i = 1}^{n} i^2 = {n(n + 1)(2n + 1) \over 6}\]

Answer 1

Hello Sir Elias, can you explain it to me in simple terms?

Answer 2

By induction. I think that I did this before on this site.

Answer 3

Is there no other direct proof?

Answer 4

I do not know another one.

Answer 5

OK. Can you help me prove it? I did it already for 1.
Now for the difficult parts: \(k\) and \(k + 1\).

Answer 6

\[(n+1)^3-n^3=3n^2+3n+1\]\[(n)^3-(n-1)^3=3(n-1)^2+3(n-1)+1\]\[(n-1)^3-(n-2)^3=3(n-2)^2+3(n-2)+1\]...
...
\[(2)^3-(1)^3=3(1)^2+3(1)+1\]add them all LHS's and RHS's

Answer 7

... @mukushla Thank you. Can you please explain?

Answer 8

we know that\[(n+1)^3-n^3=3n^2+3n+1\]

Answer 9

But, why are we doing \((n + 1)^3\) when that is completely out of subject?

Answer 10

when we add LHS's they will cancel each other

Answer 11

and because that difference of cubes generates n^2 terms for us

Answer 12

I still don't get it :(
Perhaps because I'm not a genius like you guys =D

Answer 13

no thats because i cant explain well :(

Answer 14

0 0 [1] 5 14 30 55 91 ; squares added
0 1 [4] 9 16 25 36 ; odds added euqls squares
1 3 [5] 7 9 11 ; odds
2 2 [2] 2 2 ; constants of 2s

this setup is related to pascals triangle, and when a sequence has multiple tiers of differences, till it reaches a constant, we can form a rule for it using those [..] as coeefs
\[a\frac{x^0}{0!}+b\frac{n}{1!}+c\frac{n(n-1)}{2!}+d\frac{n(n-1)(n-2)}{3!}+...\]
\[a=1;~b=4;~c=5;~d=2\]

Answer 15

ignore the x^0, its left over froma bad idea lol

Answer 16

if we turn these rows of differences clockwise 90 degrees we get the relation to the pascals triangle

1
2 1
2 3 1
2 5 4 1
2 7 9 5 1

i spose this should be flipped about so the 1s are on the other side to be consistent with the coeffs

Answer 17

and to note, my setup starts with n=0; but can be modified for n=1 by subtracting 1 from all the n parts to adjust

Answer 18

or just use the 5,9,7,2 instead :)

Answer 19

ugh, went the wrong way ... 0,1,3,2 for n=1,2,3,...

http://www.wolframalpha.com/input/?i=n%2B3n%28n-1%29%2F2%2B2n%28n-1%29%28n-2%29%2F6

Answer 20

you forgot this picture
http://jeremykun.files.wordpress.com/2011/06/triangle-proof.png

Answer 21

Oh, yeah. lol