Let P(n) be the statement that

For all postive integers, .

To prove this statement by using mathematics induction, which of the following is FALSE?

Group of answer choices

The base step is P(1).

In the inductive step, we assume P(k) is True for  and prove P(k+1) is True. 

In the inductive step, the assumption that  plays a role in proving P(k+1) is True.

In the inductive step, the induction hypothesis plays a role in proving P(k+1) is True.

To analyze which of the statements is **FALSE** in the context of a mathematical induction proof for \( P(n) \), let's go through each statement:

1. **The base step is P(1)**: This is a **true** statement. In a typical proof by induction, we start by verifying the base case, usually \( P(1) \), to confirm that the statement holds for the smallest possible value of \( n \).

2. **In the inductive step, we assume P(k) is True for \( k \geq 1 \) and prove P(k+1) is True**: This is also a **true** statement. In the inductive step, we assume that the statement holds for some arbitrary positive integer \( k \), which is called the **induction hypothesis**, and then use that to prove that the statement holds for \( k+1 \).

3. **In the inductive step, the assumption that \( k \geq 1 \) plays a role in proving P(k+1) is True**: This statement is **false**. The assumption is only that \( P(k) \) is true for some arbitrary \( k \), but we do not necessarily need to emphasize that \( k \geq 1 \) plays a specific role. The assumption that \( k \geq 1 \) is not critical in the logic of the inductive step; the focus is on the truth of \( P(k) \) rather than the value of \( k \).

4. **In the inductive step, the induction hypothesis plays a role in proving P(k+1) is True**: This is a **true** statement. The induction hypothesis \( P(k) \) is essential in proving \( P(k+1) \), as it is the foundation for demonstrating that the statement holds for the next value.

### Conclusion:
The **FALSE** statement is:
- **"In the inductive step, the assumption that \( k \geq 1 \) plays a role in proving P(k+1) is True."**

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Do you need more details on this topic or have any questions?

Here are five related questions to deepen your understanding:
1. What is the role of the base case in mathematical induction?
2. How does the inductive hypothesis help in proving the inductive step?
3. Can mathematical induction be used for proving statements about non-positive integers?
4. How would the proof change if we started the induction from a value other than 1?
5. What are some common mistakes to avoid when using mathematical induction?

**Tip**: Always ensure the inductive hypothesis is clearly stated before attempting to prove the inductive step.

Let P(n) be the statement that
For all positive integers, .
To prove this statement by using mathematics induction, which of the following is FALSE?
Group of answer choices
The base step is P(1).
In the inductive step, we assume P(k) is True for and prove P(k+1) is True.
In the inductive step, the assumption that plays a role in proving P(k+1) is True.
In the inductive step, the induction hypothesis plays a role in proving P(k+1) is True.

Explore a detailed analysis of mathematical induction, focusing on identifying a false statement in the proof process. Understand the role of the base case, inductive step, and induction hypothesis. Ideal for undergraduate students studying logic and proofs.

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