To solve the problem of finding the 300th term in the sequence described, let's first explore the behavior of the sequence:

Rules Recap:

1. Start with $3$.
2. If the current term is odd, multiply it by 3 and add 1 to get the next term.
3. If the current term is even, divide it by 2 to get the next term.

Step 1: Generating the Sequence

The sequence begins with 3:

• $3$ (odd) → $3 \times 3 + 1 = 10$
• $10$ (even) → $10 \div 2 = 5$
• $5$ (odd) → $5 \times 3 + 1 = 16$
• $16$ (even) → $16 \div 2 = 8$
• $8$ (even) → $8 \div 2 = 4$
• $4$ (even) → $4 \div 2 = 2$
• $2$ (even) → $2 \div 2 = 1$
• $1$ (odd) → $1 \times 3 + 1 = 4$ (repeats the cycle: $4 \to 2 \to 1$).

Step 2: Recognizing the Cycle

After reaching $1$, the sequence falls into a loop: $4 \to 2 \to 1$. This cycle has a length of 3 terms.

Step 3: Identifying the 300th Term

The first 7 terms before the cycle begins are $3, 10, 5, 16, 8, 4, 2$. From the 8th term onward, the cycle $4, 2, 1$ repeats.

• The position of the 300th term can be determined relative to the cycle:
  • Subtract the first 7 terms: $300 - 7 = 293$.
  • Determine the position in the cycle: $293 \mod 3 = 2$.

The 300th term corresponds to the 2nd term of the cycle, which is $2$.

Final Answer:

The 300th term in the sequence is 2.

Would you like further clarification or a breakdown of specific steps?

Expanding Understanding:

1. What is the 500th term in this sequence?
2. How long does it take for the sequence to reach the cycle starting point from any initial odd number?
3. What happens if the sequence starts with a different odd number, like 7?
4. Can we mathematically prove that any sequence starting with a positive integer will eventually reach a repeating cycle?
5. How does the sequence behave for very large initial values?

Tip: Recognizing cycles in sequences can simplify calculations and provide insights into their behavior. Look for patterns to avoid exhaustive computations!