1. The first problem asks to represent the series $(1+2) + (2+2) + (3+2) + (4+2) + (5+2)$ in sigma notation. 2. Notice each term is of the form $i + 2$ where $i$ runs from 1 to 5. 3. The sigma notation for this series is: $$\sum_{i=1}^5 (i + 2)$$ 4. The alternative option $\sum_{i=2}^5 (1 + i)$ is not equivalent because it starts at $i=2$ and the terms differ. 5. Therefore, the correct sigma notation is $\sum_{i=1}^5 (i + 2)$. 6. The second problem asks to represent the series $4^2 + 5^2 + 6^2 + 7^2$ in sigma notation. 7. Notice the terms are squares of integers starting at 4 and ending at 7. 8. We can write this as: $$\sum_{i=4}^7 i^2$$ 9. This notation sums the squares of $i$ from 4 to 7, matching the given series. Final answers: - Problem 1: $\sum_{i=1}^5 (i + 2)$ - Problem 2: $\sum_{i=4}^7 i^2$