1. **State the problem:** Express the sums using summation notation. 2. **Sum 1:** $1^3 + 2^3 + 3^3 + \dots + 9^3$ This is the sum of cubes from 1 to 9. Using summation notation: $$\sum_{k=1}^9 k^3$$ 3. **Sum 2:** $1 + 3 + 5 + 7 + \dots + (2(12) - 1)$ This is the sum of the first 12 odd numbers. The $n$th odd number is $2n - 1$. Using summation notation: $$\sum_{k=1}^{12} (2k - 1)$$ 4. **Sum 3:** $1 + 2 + 3 + 4 + \dots + 12$ This is the sum of the first 12 natural numbers. Using summation notation: $$\sum_{k=1}^{12} k$$ **Final answers:** 1) $$\sum_{k=1}^9 k^3$$ 2) $$\sum_{k=1}^{12} (2k - 1)$$ 3) $$\sum_{k=1}^{12} k$$