1. We are given two summation expressions and need to verify their correctness. 2. First, consider the summation $$\sum_{r=1}^3 (3r + 4)$$. 3. According to the summation property, this can be split as: $$\sum_{r=1}^3 (3r + 4) = \sum_{r=1}^3 3r + \sum_{r=1}^3 4$$ 4. Evaluate each summation separately: - $$\sum_{r=1}^3 3r = 3(1) + 3(2) + 3(3) = 3 + 6 + 9 = 18$$ - $$\sum_{r=1}^3 4 = 4 + 4 + 4 = 12$$ 5. Add the results: $$18 + 12 = 30$$ 6. Calculate $$\sum_{r=1}^3 (3r + 4)$$ directly as a check: $$3(1) + 4 + 3(2) + 4 + 3(3) + 4 = (3 + 4) + (6 + 4) + (9 + 4) = 7 + 10 + 13 = 30$$ 7. Both methods give the same result, so the first statement is correct. 8. Next, verify the second summation: $$\sum_{r=1}^4 (4r) = 4\sum_{r=1}^4 r$$ 9. Calculate each side: - Left side: $$4(1) + 4(2) + 4(3) + 4(4) = 4 + 8 + 12 + 16 = 40$$ - Right side: $$4 \times (1 + 2 + 3 + 4) = 4 \times 10 = 40$$ 10. Both sides equal 40, confirming the validity of the distributive property in summation. Final conclusion: Both summation expressions are correct as shown by evaluating and comparing both sides.