The 1st term in a number series is 8. From the 2nd term onwards, if the previous term is odd, then we subtract 1 from it and multiply it by 3. If the previous term is even, then we divide it by 2, before adding 2 to it. Find the 100th term in the series. 

We are given a recursive sequence where the first term is 8, and the following terms are determined by specific rules based on whether the previous term is odd or even:

1. If the previous term is **odd**, subtract 1 from it and multiply by 3.
2. If the previous term is **even**, divide it by 2 and add 2.

We are tasked with finding the 100th term in the series.

### Let's work through this step-by-step:
- \( a_1 = 8 \) (this is given)

Now, we apply the rule to generate further terms:

1. \( a_1 = 8 \) (even)  
   \( a_2 = \frac{8}{2} + 2 = 4 + 2 = 6 \)

2. \( a_2 = 6 \) (even)  
   \( a_3 = \frac{6}{2} + 2 = 3 + 2 = 5 \)

3. \( a_3 = 5 \) (odd)  
   \( a_4 = (5 - 1) \times 3 = 4 \times 3 = 12 \)

4. \( a_4 = 12 \) (even)  
   \( a_5 = \frac{12}{2} + 2 = 6 + 2 = 8 \)

Now we see that \( a_5 = a_1 \), meaning that the sequence repeats every 4 terms:  
\[ 8, 6, 5, 12 \]

### To find the 100th term, we observe that:
- The sequence repeats every 4 terms.
- So, the position of the 100th term within the cycle is given by \( 100 \mod 4 = 0 \).

This means the 100th term is the same as the 4th term in the sequence, which is 12.

### Conclusion:
The 100th term in the series is \( \boxed{12} \).

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Would you like more details on how I derived the cyclic pattern? Here are some related questions:

1. What is the 200th term in the series?
2. How can we generalize this method for any term in the series?
3. Can you explain the behavior of sequences with similar rules?
4. What other sequences involve alternating rules based on odd/even conditions?
5. How can we prove the periodicity of the sequence mathematically?

**Tip:** When you encounter recursive sequences with defined conditions, always check for patterns or cycles after a few iterations.

The 1st term in a number series is 8. From the 2nd term onwards, if the previous term is odd, then we subtract 1 from it and multiply it by 3. If the previous term is even, then we divide it by 2, before adding 2 to it. Find the 100th term in the series.

Learn how to solve a recursive sequence problem where the first term is 8, and the next terms follow a set of rules based on whether the previous term is odd or even. Discover the 100th term using a step-by-step explanation. Suitable for Grades 9-12.

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