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?

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.