Summation Problems

Question

How did this particular equation come about? I haven't seen it before in the summation rules index on wikipedia:

$$\sum\limits_{i=1}^{k+1} x_i =\left(\sum\limits_{i=1}^{k} x_i\right)+x_{k+1} $$

Edit: I guess what i meant to say was if there was any sort of proof for this equation.

Answer 1

Since $$\sum\limits_{i=1}^{k} x_i =x_1 +x_2+\cdots + x_k$$ Then $$\sum\limits_{i=1}^{\color{red}{k+1}} x_i =x_1 +x_2+\cdots + x_k+\color{red}{x_{k+1}}$$ Therefore $$\sum\limits_{i=1}^{\color{red}{k+1}} x_i = \left(\sum\limits_{i=1}^{k} x_i\right)+\color{red}{x_{k+1}}$$

Answer 2

This is a rather straightforward statement:

The sum of the first $k$ terms of a sequence, plus the "$k+1$"st term, is equal to the sum of the first $k+1$ terms of this sequence.

Try not to let complicated-looking notation get in the way of your understanding!