1. **Problem statement:** Prove the summation properties: $$\sum_{k=1}^n (x_k + y_k) = \sum_{k=1}^n x_k + \sum_{k=1}^n y_k$$ $$\sum_{k=1}^n (a x_k) = a \sum_{k=1}^n x_k$$ $$\sum_{k=1}^n x_k = \sum_{k=1}^m x_k + \sum_{k=m+1}^n x_k$$ 2. **Formula and explanation:** The summation operator $\sum$ is linear, meaning it distributes over addition and scalar multiplication. 3. **Proof of first property:** $$\sum_{k=1}^n (x_k + y_k) = (x_1 + y_1) + (x_2 + y_2) + \cdots + (x_n + y_n)$$ $$= (x_1 + x_2 + \cdots + x_n) + (y_1 + y_2 + \cdots + y_n) = \sum_{k=1}^n x_k + \sum_{k=1}^n y_k$$ 4. **Proof of second property:** $$\sum_{k=1}^n (a x_k) = a x_1 + a x_2 + \cdots + a x_n = a (x_1 + x_2 + \cdots + x_n) = a \sum_{k=1}^n x_k$$ 5. **Proof of third property:** $$\sum_{k=1}^n x_k = x_1 + x_2 + \cdots + x_m + x_{m+1} + \cdots + x_n$$ $$= (x_1 + x_2 + \cdots + x_m) + (x_{m+1} + \cdots + x_n) = \sum_{k=1}^m x_k + \sum_{k=m+1}^n x_k$$