# Thread: Examples of Identity of Finite Sum of Integers

1. ## Examples of Identity of Finite Sum of Integers

Hello.

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

Thank you.

Misako Kawasoe

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

3. Originally Posted by chiph588@
How about $\displaystyle \sum_{i=1}^{n} i{n \choose i} = n2^{n-1}$.
How about $\displaystyle \sum_{j=1}^{n}j^2{j\choose n}=n(2n+1)2^{n-2}$?

4. $\displaystyle 3^2+4^2=5^2$

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

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

5. Originally Posted by Drexel28
How about $\displaystyle \sum_{j=1}^{n}j^2{j\choose n}=n(2n+1)2^{n-2}$?
Or... $\displaystyle \sum_{j=1}^{n}j^3{j\choose n}=n^2(n+3)2^{n-3}$

6. Originally Posted by chiph588@
Or... $\displaystyle \sum_{j=1}^{n}j^3{j\choose n}=n^2(n+3)2^{n-3}$
Haha. Or! Define $\displaystyle f_1(x)=n(1+x)^{n-1}$ and define $\displaystyle f_{m+1}(x)=\left(x\cdot f_m(x)\right)'$. Then, $\displaystyle \sum_{j=1}^{n}j^m{n\choose j}=f_m(1),\text{ }m\geqslant 1$

7. 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 $\displaystyle 70^2 = 1^2 + 2^2 + \dots + 24^2$ without using sigma symbol...
Anyway, thanks a lot!