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To identify the inductive hypothesis, let's first analyze the given statement \( P(n) \):

The statement \( P(n) \) asserts that:
\[
\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{n \cdot (n+1)} = \frac{n}{n+1}.
\]

In mathematical induction, the inductive hypothesis assumes the statement is true for some integer \( k \). Thus, we assume:
\[
P(k): \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{k \cdot (k+1)} = \frac{k}{k+1}.
\]

Looking at the options provided:
1. The first choice suggests we assume \( P(k) \) with the expression equal to \( \frac{k+1}{k+2} \).
2. The second option suggests \( P(k) \) with the expression equal to \( \frac{k-1}{k+1} \).
3. The third option matches \( P(k) \) with the expression \( \frac{k}{k+1} \), which is consistent with the form of the original statement \( P(n) \).
4. The fourth option suggests the form \( \frac{k-1}{k+1} \), which is incorrect as it does not align with the structure of \( P(n) \).

The correct answer, therefore, is:
**The third option**: "In the inductive hypothesis, we assume \( P(k) \) for some integer \( k > 0 \) and that \( \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{k \cdot (k+1)} = \frac{k}{k+1} \)."

Would you like more details on this or have any questions?

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Here are five related questions:
1. How do you establish the base case in an inductive proof?
2. What is the next step after assuming the inductive hypothesis?
3. How would you prove \( P(k+1) \) based on \( P(k) \)?
4. Can this method of proof apply to non-integer sequences?
5. What are common pitfalls in formulating the inductive hypothesis?

**Tip:** In mathematical induction, always check that your inductive hypothesis aligns exactly with the form of the statement being proven.

Identify the inductive hypothesis for the statement that \( \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{n \cdot (n+1)} = \frac{n}{n+1} \).

Learn how to identify the inductive hypothesis for the series \( \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{n \cdot (n+1)} = \frac{n}{n+1} \). This problem involves key concepts in mathematical induction and sequence summation, suitable for advanced high school students.

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