Examples of Identity of Finite Sum of Integers

**k-misako** · April 6th 2010, 06:44 AM

Hello.

If someone know interesting examples of identity of finite sum of integers such as $\textstyle 70^2 = \sum_{k=1}^{24} k^2$ , could you tell me some please.

Thank you.

Misako Kawasoe

**chiph588@** · April 6th 2010, 07:28 AM

How about $\sum_{i=1}^{n} i{n \choose i} = n2^{n-1}$ .

**Drexel28** · April 6th 2010, 07:59 AM

Originally Posted by chiph588@

How about $\sum_{i=1}^{n} i{n \choose i} = n2^{n-1}$ .

How about $\sum_{j=1}^{n}j^2{j\choose n}=n(2n+1)2^{n-2}$ ?

**Bruno J.** · April 6th 2010, 08:10 AM

$3^2+4^2=5^2$

$3^3+4^3+5^3=6^3$

$1^3+2^3+\dots+n^3=(1+2+\dots+n)^2$

**chiph588@** · April 6th 2010, 09:44 AM

Originally Posted by Drexel28

How about $\sum_{j=1}^{n}j^2{j\choose n}=n(2n+1)2^{n-2}$ ?

Or... $\sum_{j=1}^{n}j^3{j\choose n}=n^2(n+3)2^{n-3}$

**Drexel28** · April 6th 2010, 11:43 AM

Originally Posted by chiph588@

Or... $\sum_{j=1}^{n}j^3{j\choose n}=n^2(n+3)2^{n-3}$

Haha. Or! Define $f_1(x)=n(1+x)^{n-1}$ and define $f_{m+1}(x)=\left(x\cdot f_m(x)\right)'$ . Then, $\sum_{j=1}^{n}j^m{n\choose j}=f_m(1),\text{ }m\geqslant 1$

**k-misako** · April 11th 2010, 04:59 AM

Thank you so much, everyone!
I really appreciate these replies.
The Bruno J.-san's example fits my request best because I want a "particular" example rather than general one. I should have written in my first post $70^2 = 1^2 + 2^2 + \dots + 24^2$ without using sigma symbol...
Anyway, thanks a lot!

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