Splitting Summation

Question

While studying for my course, I found following rule in the web (below image). My professor seems to assume everyone know this, but I don't understand why this works. It says $$1 + 2 + 3 + \cdots + n= [ 1 + 2 + 3 + \cdots + n/2 ] + [n/2 + \frac{n+1}{2} + \cdots + n]$$
Can you explain how this summation rule works?

$$\large{}\sum_{k=1}^n k= \sum_{k=1}^{\frac{n}{2}} k + \sum_{k=\frac{n}{2}+1}^n k$$

I have trouble understanding the confusion. Are you wondering about what happens when $n$ is not even, so $n/2$ is not an integer? — Joshua P. Swanson, Sep 17 '20 at 06:51

score 0 · Answer 1 · answered Sep 17 '20 at 05:46

It's just simply splitting the original sum into two parts.

$\sum_{k=1}^n k\\ =1+2+\cdots+n\\ =(1+2+\cdots+\frac{n}{2})+((\frac{n}{2}+1)+(\frac{n}{2}+2)+\cdots+n)\\ =\sum_{k=1}^{n/2} k +\sum_{k=n/2+1}^n k$

For example, if $n=10$ :

$1+2+3+4+5+6+7+8+9+10\\ =(1+2+3+4+5)+(6+7+8+9+10)$

Splitting Summation

1 Answers1