1. **Stating the problem**: We have a sequence starting at 3. Each term is generated from the previous as follows: - If the term is even, next term = term \div 2 - If the term is odd, next term = 3 \times term + 1 We want to find the 300th term. 2. **Generate the sequence starting from 3**: Term 1: 3 (given) Term 2: 3 is odd, so 3 \times 3 + 1 = 10 Term 3: 10 is even, so 10 \div 2 = 5 Term 4: 5 is odd, so 3 \times 5 + 1 = 16 Term 5: 16 is even, so 16 \div 2 = 8 Term 6: 8 is even, so 8 \div 2 = 4 Term 7: 4 is even, so 4 \div 2 = 2 Term 8: 2 is even, so 2 \div 2 = 1 Term 9: 1 is odd, so 3 \times 1 + 1 = 4 Term 10: 4 is even, so 4 \div 2 = 2 Term 11: 2 is even, so 2 \div 2 = 1 3. **Identify the pattern**: From term 8 onwards, the sequence cycles through 1,4,2,1,4,2,... This 3-term cycle repeats forever. 4. **Find the 300th term**: The cycle length is 3, starting at term 8. Calculate the offset of term 300 from term 8: 300 - 8 + 1 = 293 terms into the cycle. We find the position inside the cycle by computing remainder when dividing 293 by 3: $$293 \div 3 = 97\text{ remainder } 2$$ - If remainder 1: term is 1 - If remainder 2: term is 4 - If remainder 0: term is 2 Here remainder is 2, so the 300th term is 4. **Final answer:** $$\boxed{4}$$