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Can anyone explain to me how you would derive this ? I have this question asked in a CS class and can't figure out how to derive it. it has to be derived as you would with sum of N


1 2    3  ......  N 
N N-1  N-2    ....1 
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What do you want? $\sum_{N=1}^k N^2 =\text{?} $ – draks ... Mar 13 '14 at 6:39
yes, how to derive the closed form of that sum – Tangleman Mar 13 '14 at 6:41

3 Answers 3

Okay, someone will post a method of common differences soon enough, so let's take a new approach. Combinatorics. Particularly because I recently learnt this myself.

Consider this: How many ways can I choose ordered triples $(a,b,c)$ from $0\le a,b\lt c\le n$?

For fixed $c$ this can be done in $c^2$ ways, because $a$ and $b$ can independently take values in the set $\{0,1,2,\cdots,c-1\}$. Since $c$ can take any value between $1$ and $n$, the total number of ways is $$1^2+2^2+\cdots+n^2$$

Now to find this number combinatorically!

There are $C(n+1,2)$ triples of the form $(a,a,c)$. To form triples of the form $(a,b,c)$ with $a\ne b$ We can select $a,b,c$ in $C(n+1,3)$ ways, and to each way there are two triples, $(a,b,c)$ and $(b,a,c)$.

Thus we can conclude that $$1^2+2^2+\cdots+n^2 = {n+1\choose 2}+2{n+1\choose 3}$$

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+1 nice. related: – draks ... Mar 13 '14 at 7:07
@draks... that is my question. I did say I recently learnt this myself – Sabyasachi Mar 13 '14 at 7:10
no offense. why not linking it? – draks ... Mar 13 '14 at 7:11
@draks... fair enough. I will. – Sabyasachi Mar 13 '14 at 7:12
@Sabyasachi you mentioned in the other link that you can derive this using method of common differences, do you know how to do it that way? – Tangleman Mar 13 '14 at 18:24

Here's my favourite trick for $\sum_{k=1}^N k^2$. Note that $(k+1)^3 - (k-1)^3 = 6 k^2 + 2$. So $$\sum_{k=1}^N \left((k+1)^3 - (k-1)^3\right) = \sum_{k=1}^N (6 k^2+2)$$ Now if you look closer at the sum on the left, you see a lot of cancellations: all the cubes from $2^3$ to $(N+1)^3$ are there with $+$ signs, and all those from $0^3$ to $(N-1)^3$ are there with $-$ signs. All that's left after cancellation is $N^3 + (N+1)^3 - 0^3 - 1^3 = N^3 + (N+1)^3 - 1$. On the right, we have $6 \sum_{k=1}^N k^2+ \sum_{k=1}^N 2 = 2 N + 6 \sum_{k=1}^N k^2$. Subtract $2N$ from both sides, divide by $6$ and simplify...

You can get a formula for $\sum_{k=1}^N k^3$ similarly, starting with $(k+1)^4 - (k-1)^4 = 8 k^3 + 8 k$.

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absolute magic :D – Sabyasachi Mar 13 '14 at 7:14

The result is a polynomial of third degree $ak^3+bk^2+cx+d$. Collect four examples $k=1,2,3,4$, get the coefficients $a,b,c,d$ and use proof by induction.

Good luck,

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I am really frustrated because the professor said he doesn't want to see induction! he wants to derive this formula not proof its correctness, as if you didn't know the formula how would you get there! he gave us example of sum of N but N^2 is a whole different monster – Tangleman Mar 13 '14 at 6:47
@Tangleman does combinatorics work for you? see my answer. – Sabyasachi Mar 13 '14 at 6:52
@Sabyasachi thanks, that does look like something he is asking but I don't quit understand the whole thing. I will try to study it more and see if I can get the jest of it. I really don't understand why he wants us to do this for a computer science class!! – Tangleman Mar 13 '14 at 6:58
@Tangleman being good at math(and combinatorics) will help you in computational complexity analysis. – Sabyasachi Mar 13 '14 at 7:01

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