To solve this question, we are working on proving a statement $P(n)$ using mathematical induction. The statement $P(n)$ is given as:

$P(n): \quad \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{n \cdot (n+1)} = \frac{n}{n+1}$

For the basis step of mathematical induction, we need to show that $P(1)$ is true. This will involve substituting $n = 1$ into the statement and checking if it holds.

Step-by-Step Solution

1. Basis Step $P(1)$: Substitute $n = 1$ in the equation: $\frac{1}{1 \cdot 2} = \frac{1}{2}$ According to $P(1)$, we expect this to be equal to $\frac{1}{1+1} = \frac{1}{2}$. So, $P(1)$ is true because both sides of the equation are equal.

2. Examining Answer Choices:

   • $P(1)$ is true: This is correct based on our calculation.
   • $P(0)$ is true: There is no information or basis for a $P(0)$ in this context, as our sequence starts with $n = 1$.
   • $P(1) \rightarrow P(2)$ is true: This would be part of the inductive step, which we are not required to check in the basis step. For the basis step, we only need $P(1)$.
   • $\frac{1}{2} = \frac{1}{2}$: This is true, but it is a generic statement that doesn't address the specific basis of $P(1)$.
   • $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{n \cdot (n+1)} = \frac{n}{n+1}$ is true for $n = 1$: This is the correct basis step statement, and it is true as we verified.
   • $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{n \cdot (n+1)} = \frac{n}{n+1}$ is true for some integer $n$: This statement is vague and does not focus on the basis step specifically.

Conclusion

For the basis step, we should select:

• $P(1)$ is true
• $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{n \cdot (n+1)} = \frac{n}{n+1}$ is true for $n = 1$

These are the appropriate selections to complete the basis step.

Do you have any questions or need further clarification?

Related Questions:

1. How would you proceed with the inductive step for proving $P(n)$ for all $n$?
2. Why is $P(0)$ not relevant in this context?
3. Can you explain why the basis step is critical in mathematical induction?
4. How would the proof change if the series started from $n = 2$ instead?
5. What would happen if $P(1)$ was not true?

Tip:

When using induction, always clearly separate the basis step (initial case) from the inductive step (assuming $P(k)$ is true to prove $P(k+1)$).