1. **Problem statement:** Find the sums of the given arithmetic series using the provided formulas and verify the answers. 2. **Part (a): Sum of 1 + 2 + 3 + ... + 999** - Number of terms: $n = 999$ - Average of first and last term: $\frac{1 + 999}{2} = 500$ - Sum formula: $$S = n \times \text{average} = 999 \times 500 = 499500$$ - **Answer:** $499500$ 3. **Part (b): Sum of 3 + 6 + 9 + ... + 582** - Number of terms: $\frac{582}{3} = 194$ - Average of first and last term: $\frac{3 + 582}{2} = \frac{585}{2} = 292.5$ - Sum formula: $$S = \text{number of terms} \times \text{average} = 194 \times 292.5 = 56745$$ - **Answer:** $56745$ 4. **Part (c)(i): Sum of 5 + 10 + 15 + ... + 1550** - Number of terms: $\frac{1550}{5} = 310$ - Average of first and last term: $\frac{5 + 1550}{2} = \frac{1555}{2} = 777.5$ - Sum formula: $$S = 310 \times 777.5 = 241025$$ - **Answer:** $241025$ 5. **Part (c)(ii): Sum of 12 + 24 + 36 + ... + 1104** - Number of terms: $\frac{1104}{12} = 92$ - Average of first and last term: $\frac{12 + 1104}{2} = \frac{1116}{2} = 558$ - Sum formula: $$S = 92 \times 558 = 51336$$ - **Answer:** $51336$ 6. **Sum of odd numbers series 1 + 3 + 5 + ... + 733** - Number of terms: $\frac{733 + 1}{2} = 367$ - Average: $\frac{n + 1}{2} = 367$ - Sum formula: $$S = \text{number of terms}^2 = 367^2 = 134689$$ - **Answer:** $134689$ All answers match the provided values, confirming correctness.