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#1 2006-03-13 01:32:48

George,Y
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Registered: 2006-03-12
Posts: 1,306

1²+2²+3²+...+n²=? without using induction

the answer seems too odd for me -n(n+1)(2n+1)/6, and what mathematical induction can do is just to prove other than to derive. Hoping some genius could handle this out, with ease and simplisity, like the derivation of 1+a+a²+a³+...+an

it's a huge challenge!! Cuz Google doesn't provide a derivation!!:P


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#2 2006-03-13 01:42:34

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

Re: 1²+2²+3²+...+n²=? without using induction

I don't see why you don't want it to be using induction, but whatever. I remember that question being asked here before, so there'll be a topic here somewhere with a proof. I remember that it has an induction proof and a different proof as well, so you get two answers!

I'll try to find the link now. Hang on.

Edit: That was easier than I thought! Here it is.

Last edited by mathsyperson (2006-03-13 01:43:42)


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It wanted to be normal.

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#3 2006-03-13 01:48:48

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

Re: 1²+2²+3²+...+n²=? without using induction

Many thanks! roll smile


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#4 2006-03-13 15:53:32

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

Re: 1²+2²+3²+...+n²=? without using induction

Thank you for the link. The link finally reminded me of the proof i once read in Thomas' Calculus book, it uses a neighbor elimation trick:
key thought --  (k+1)³-k³ =3k²+3k+1

     3(1²+2²+3²+...+n²)
     3(1 +2 +3 +... +n)
+)     1 +1 +1 +... +1
___________________

(-1³+2³)+(-2³+3³)+(-3³+4³)+...+(-n³+(n+1)³)
=-1+(n+1)³=n³+3n²+3n

3x + 3n(n+1)/2+n=n³+3n²+3n

3x=n³+3/2 n²+1/2 n= 1/2 n(2n²+3n+1)

x= 1/6 n (2n+1) (n+1) cool


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#5 2006-03-18 03:31:06

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: 1²+2²+3²+...+n²=? without using induction

Hi.
The function:


is very useful for calculating sums.
That's because:

If you want to calculate this:

, you may express the f(x) = x^i as:
,
where r(x) is the "remainder", which has less power than f(x).
Example for f(x)=x^1:
S_2 (x) = (x+1)^2-x^2=x^2+2x+1-x^2=2x+1
(S_2 (x))/2=x+1/2
f(x)=x^1=x=0.5 S_2(x) - 0.5 = 0.5(S_2(x)-1).
So
.

There's also and binomial proof, which is more usable and universal, but it's harder too.


IPBLE:  Increasing Performance By Lowering Expectations.

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#6 2006-03-19 14:45:22

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

Re: 1²+2²+3²+...+n²=? without using induction

Good proof! Thanks a lot

a small error, though
f(x) = x^i should be x^k instead, same to the following

in order to know sum of i^k, we need to do know k-1 sums of i^l ,for l=1,2,...k-1
but to me, sum of squares is enough to derive Spearman's Coefficient tongue

Last edited by George,Y (2006-03-19 14:51:28)


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#7 2006-03-31 07:45:05

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: 1²+2²+3²+...+n²=? without using induction

Yes, thank you.


IPBLE:  Increasing Performance By Lowering Expectations.

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