Suppose you know the following about a statement P(n)
.

P(4)
, P(8)
 and P(12)
 are all true.
P(1)
 and P(3)
 are false.
For all integers k≥5
, if P(k)
 is true then P(k+1)
 is true.
What is the smallest integer x
 for which you can be sure that P(n)
 is true for all integers  n≥

Given the information:

1. \( P(4) \), \( P(8) \), and \( P(12) \) are true.
2. \( P(1) \) and \( P(3) \) are false.
3. For all integers \( k \geq 5 \), if \( P(k) \) is true, then \( P(k+1) \) is true.

### Goal
We need to find the smallest integer \( x \) such that \( P(n) \) is true for all \( n \geq x \).

### Analysis

1. **Starting at \( P(4) \)**:
   - \( P(4) \) is true, but this doesn't necessarily tell us about \( P(5) \) because the implication rule \( P(k) \rightarrow P(k+1) \) only applies starting from \( k \geq 5 \).

2. **Considering \( P(8) \) and beyond**:
   - \( P(8) \) is true.
   - By the rule \( P(k) \rightarrow P(k+1) \) for \( k \geq 5 \), we know \( P(9) \) is also true.
   - This rule continues, implying \( P(10) \), \( P(11) \), and so on are true as well.

3. **Considering \( P(12) \) and beyond**:
   - \( P(12) \) is already known to be true, and again, by the implication rule, this ensures that all subsequent \( P(n) \) for \( n \geq 12 \) are true.

### Conclusion

- Since \( P(8) \) is true and \( P(k) \) implies \( P(k+1) \) for \( k \geq 5 \), we can conclude that \( P(n) \) is true for all \( n \geq 8 \).
- Therefore, the smallest integer \( x \) for which you can be sure that \( P(n) \) is true for all \( n \geq x \) is **8**.

### Answer:
The smallest integer \( x \) is **8**.

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Would you like more information or have any questions about this? Here are some related questions that might be helpful:

1. How does the principle of mathematical induction relate to statements like this?
2. What role does the base case play in induction-like reasoning?
3. How can logical implications help in proving statements for large sets of numbers?
4. What strategies can be used when determining the truth of recursive statements?
5. How does one determine the validity of a statement when only partial information is known?

**Tip:** In problems involving logical implications and sequences, identifying the point where an implication begins and then applying it iteratively is crucial to solving for the general case.

Explore the application of mathematical induction and recursive statements through a detailed example. Learn how to prove the truth of statements for large sets of numbers using logical implications and base cases. Suitable for Grades 10-12 students.

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