The first term in a sequence is . We can determine the next terms in the sequence as follows:

If a term is even, then divide it by  to get the next term.

If a term is odd, then multiply it by  and add  to get the next term.

By doing this, we can determine that the first three terms in the sequence are   and 

Elias writes the first  n terms in this sequence and notices that the sum of these terms is a four-digit number. What is the smallest possible value of n?

We are given a sequence where the first term is 5. The rules for generating the sequence are as follows:

- If a term is **even**, divide it by 2.
- If a term is **odd**, multiply it by 3 and add 1.

The first three terms are provided: 5, 16, and 8. Let's generate more terms following these rules, and find the smallest \( n \) such that the sum of the first \( n \) terms is a four-digit number.

### Generating the Sequence:
1. The first term is 5.
    - Since 5 is odd, we apply the rule \( 3 \times 5 + 1 = 16 \).
2. The second term is 16.
    - Since 16 is even, we divide by 2 to get \( 16 / 2 = 8 \).
3. The third term is 8.
    - Since 8 is even, divide by 2 to get \( 8 / 2 = 4 \).
4. The fourth term is 4.
    - Since 4 is even, divide by 2 to get \( 4 / 2 = 2 \).
5. The fifth term is 2.
    - Since 2 is even, divide by 2 to get \( 2 / 2 = 1 \).
6. The sixth term is 1.
    - Since 1 is odd, apply the rule \( 3 \times 1 + 1 = 4 \).

At this point, we see that the sequence starts to repeat (since we reached 4 again, which will generate the same terms we already saw).

### Sequence: 
\[
5, 16, 8, 4, 2, 1, 4, 2, 1, \dots
\]

This is a repeating cycle starting at the 4th term.

### Calculating the Sum:

We now calculate the sum of the sequence and determine the smallest \( n \) for which the sum is a four-digit number.

The sum of the first 6 terms:
\[
5 + 16 + 8 + 4 + 2 + 1 = 36
\]

Since the sequence starts repeating after the 6th term, the next terms are \( 4, 2, 1 \) repeating in a cycle. The sum of one cycle (terms 4, 2, and 1) is:
\[
4 + 2 + 1 = 7
\]

Now, let's continue adding cycles to reach a sum of at least 1000.

#### Sum after \( n = 6 \) terms: 
\[
36
\]

Adding the next cycle of \( 4 + 2 + 1 \):
\[
36 + 7 = 43 \quad (n = 9)
\]
Adding another cycle:
\[
43 + 7 = 50 \quad (n = 12)
\]
Keep adding cycles until we reach a sum of 1000:

\[
\text{Sum for \( n = 144 \) terms: } 36 + (7 \times   terms\]= Total four digits} term \( cycle qusts during four Digits}.

The first term in a sequence is 5. We can determine the next terms in the sequence as follows:
If a term is even, then divide it by 2 to get the next term.
If a term is odd, then multiply it by 3 and add 1 to get the next term.
By doing this, we can determine that the first three terms in the sequence are 5, 16, and 8.
Elias writes the first n terms in this sequence and notices that the sum of these terms is a four-digit number. What is the smallest possible value of n?

Explore the Collatz Conjecture sequence starting with 5. Learn how to find the smallest value of n such that the sum of the first n terms results in a four-digit number. This problem introduces key concepts in number sequences and summation. Suitable for high school students in Grades 9-12.

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