How to properly solve this two summation sitting next with each other?

Question

I just want to ask how do we solve this kind of summation?

$$\sum_{i=1}^n x_i\sum_{i=1}^n y_i$$

I am confused how

• Do we solve for each summation first? then multiply the summation of x, to the summation of y?
• Or do we solve the summation of all $y_i$ first, then we multiply it to each element of $x_i$ changing all the value of $x_i$, and then only after that, we get the summation of the $x_i$?

Answer 1

They are equivalent: $$(x_1+x_2)(y_1+y_2) = x_1(y_1+y_2) + x_2(y_1+y_2)$$

Answer 2

\begin{align}\left(\sum_{i=1}^m x_i\right)\left(\sum_{j=1}^n y_i\right)&=\sum_{i=1}^m\sum_{j=1}^nx_iy_j\\ &=\sum_{i=1}^m\left(x_i\sum_{j=1}^ny_j\right) \end{align}