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 August 14th, 2014, 05:50 PM #1 Senior Member     Joined: Feb 2014 From: Louisiana Posts: 156 Thanks: 6 Math Focus: algebra and the calculus Deriving the formula for summing a sequence of squares If we have the sequence $\displaystyle 1^{2}, 2^{2}, 3^{2}, 4^{2},\cdots n^{2}$, how do we derive $\displaystyle \sum_{k = 1}^{n}k^{2}$ for any positive integer $\displaystyle n$?
 August 14th, 2014, 06:01 PM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms The general solution for this problem is complex and involves Bernoulli numbers. But if you know that the sum of an n-th degree polynomial is an (n+1)-th degree polynomial, you can do a simple curve fit through the points (0, 0), (1, 1), and (2, 5).
 August 14th, 2014, 06:09 PM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 6,966 Thanks: 2283 Math Focus: Mainly analysis and algebra Hi Mr. Davis, I think you posted a question on sequences for which the answer was $n(n+1)$. The method of differences that I used there will serve you well here too.
 August 14th, 2014, 07:38 PM #4 Senior Member   Joined: Sep 2012 From: British Columbia, Canada Posts: 764 Thanks: 53 Before you post a question, take some time to search for it, in case it was already answered. In this case, WikiProof gives a nice proof of it.
August 14th, 2014, 08:39 PM   #5
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Quote:
 Originally Posted by CRGreathouse But if you know that the sum of an n-th degree polynomial is an (n+1)-th degree polynomial, you can do a simple curve fit through the points (0, 0), (1, 1), and (2, 5). $\displaystyle \ \ \ \$ <----- Did you mean (2, 4)?
But the third degree polynomial will require four different points for curve-fitting.

 August 14th, 2014, 08:53 PM #6 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms I did not mean (2, 4), I meant (2, 5). But you're right, you need another point. Take (3, 14).
August 14th, 2014, 10:01 PM   #7
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Quote:
 Originally Posted by CRGreathouse I did not mean (2, 4), I meant (2, 5). But you're right, you need another point. Take (3, 14).
I asked about (2, 4), because the OP's question deals with the sum of squares.

In keeping with that, I expected you to give (0, 0), (1, 1), (2, 4), ...

 August 15th, 2014, 03:29 AM #8 Math Team   Joined: Apr 2010 Posts: 2,778 Thanks: 361 The question is indeed about the sum of the (first lowest) n squares, which are $\displaystyle 0, 1^2 = 1, 1^2 + 2^2 = 5, 1^2 + 2^2 + 3^2 = 14,...$. To find them for some n I'd write $\displaystyle k^2 = 2 \cdot {k \choose 2} + {k \choose 1}$ and then $\displaystyle \sum_{k=1}^{n}k^2 = \sum_{k=1}^{n}\left(2 \cdot {k \choose 2} + {k \choose 1}\right) = 2 \cdot {n+1 \choose 3} + {n+1 \choose 2}$
 August 15th, 2014, 03:22 PM #9 Global Moderator   Joined: Dec 2006 Posts: 18,034 Thanks: 1393 $\displaystyle \text{As }6k^2 \equiv 2(k^3 - (k - 1)^3) + 3(k^2 - (k - 1)^2) + 1,$ $\displaystyle 6\sum_{k=1}^n k^2 = \sum_{k=1}^n (2(k^3 - (k - 1)^3) + 3(k^2 - (k - 1)^2) + 1) = 2n^3 + 3n^2 + n = n(n + 1)(2n + 1).$
August 16th, 2014, 05:02 PM   #10
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Hello, Mr Davis 97!

I'll show you a very primitive solution to this problem.

Quote:
 If we have the sequence $\displaystyle 1^2, 2^2, 3^2, 4^2,\cdots n^2$, how do we derive $\displaystyle \sum_{k = 1}^{n}k^{2}$ for any positive integer $\displaystyle n$?

We have: $\:S_n \;=\;1^2+2^2+3^2+4^2+\cdots + n^2$

Crank out the first few partial sums.

$\qquad\begin{array}{ccccccc}S_1 &=& 1^2 &=& 1 \\ S_2 &=& 1^2+2^2 &=& 5 \\\ S_3 &=& 1^2+2^2+3^2 &=& 14 \\ S_4 &=& 1^2+2^2+3^2+4^2 &=& 30 \\ S_5 &=& 1^2+2^3+3^2+4^2+5^2 &=& 55 \end{array}$

Take the differences of consecutive terms,
$\quad$then take the differences of the differences, and so on.

$\begin{array}{c|ccccccccccc} S_n & 1 && 5 && 14 && 30 && 55 && 91 \\ \hline \text{1st diff} && 4 && 9 && 16 && 25 && 36 \\ \text{2nd diff} &&& 5 && 7 && 9 && 11 \\ \text{3rd diff} &&&& 2 && 2 && 2 \end{array}$

The 3rd differences are constant.
Hence, the generating function is of the 3rd degree (a cubic).

The general cubic is: $\:f(n) \:=\:an^3 + bn^2 + cn + d$

Use the first four values from our list:

$\quad \begin{array}{ccccc}f(1) \:=\:1\!: & a + b + c + d &=& 1 \\ f(2) \:=\:5\!: & 8a + 4b + 2c + d &=& 5 \\ f(3) \:=\:14\!: & 27a + 9b + 3c + d &=& 14 \\ f(4)\:=\:30\!: & 64a + 16b + 4c + d &=& 30 \end{array}$

Solve the system: $\:a = \frac{1}{3},\;b = \frac{1}{2},\;c = \frac{1}{6},\;d = 0$

Hence: $\:f(n) \;=\;\frac{1}{3}n^3 + \frac{1}{2}n^2 + \frac{1}{6}n \;=\;\frac{1}{6}n(2n^2 + 3n + 1)$

Therefore: $\displaystyle\:\sum^n_{k=1} k^2 \;=\;\frac{n(n+1)(2n+1)}{6}$

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