Question regarding Sum Notation in the least squares formula

Question

I'm attempting to figure out the difference between Σx^2 and (Σx)^2 in this least squares regression formula https://i.stack.imgur.com/bcCow.jpg. Any ideas? I figure there must be a difference.

Answer 1

For example, let $x_i$ represent $i^{th}$ natural number, and n = 4.

Then $\left(\sum_{i=1}^n x_i\right)^2$ = (1 + 2 + 3 + 4)^2 = (10)^2 = 100

and $\left(\sum_{i=1}^n x_i^2\right)$ = (1^2 + 2^2 + 3^2 + 4^2) = (1 + 4 + 9 + 16) = 30.

Answer 2

$$\sum_{i=1}^n x_i^2=x_1^2+x_2^2+\cdots+x_n^2$$ $$\Big(\sum_{i=1}^n x_i\Big)^2=(x_1+x_2+\cdots+x_n)^2$$

Answer 3

$$\left(\sum x_i\right)^2 = \sum x_i^2 + 2\sum_{i<j}x_i x_j$$