## 1.21 Theorems

1. $\sum_{i=1}^n bx_i=b \sum_{i=1}^n x_i$
2. $\sum_{i=1}^n (ax_i-b)=a \sum_{i=1}^n x_i-nb$
3. $\sum_{i=1}^n (ax_i^2-bx_i+c)=a\sum_{i=1}^n x_i^2-b\sum_{i=1}^n x_i + nc$
4. $\sum_{i=1}^n (ax_i-by_i)=a\sum_{i=1}^n x_i - b \sum_{i=1}^n y_i$
5. $\sum_{i=1}^n (ax_i-b)^2=a^2 \sum_{i=1}^n x_i^2 - 2ab \sum_{i=1}^n x_i + nb^2$
6. $(\sum_{i=1}^n x_i)^2=\sum_{i=1}^n x_i^2 + \sum_{i \ne j}^n\sum x_ix_j$
7. $\prod_{i=1}^k x_iy_i = (\prod_{i=1}^k x_i)(\prod_{i=1}^k y_i)$
8. $\sum_{i=1}^m \sum_{i=1}^n (x_i+y_j)=n\sum_{i=1}^m x_i + m \sum_{i=1}^n y_j$
9. m$$\sum_{i=1}^m \sum_{i=1}^n (x_iy_j)=(\sum_{i=1}^n x_i) (\sum_{i=1}^n y_j)$$