To prove that the sequence $a_n$ is monotonic using mathematical induction, we need to show that the sequence is either increasing or decreasing. We can do this by proving that if $a_k$ satisfies the monotonicity condition, then $a_{k+1}$ continues to satisfy the same condition.

The recursive relation for the sequence is:

$a_1 = 3, \quad a_{n+1} = \frac{1}{6 - a_n}$

Step-by-step reasoning:

1. Base Case: You will first show that the sequence is either increasing or decreasing for $a_1$.

2. Inductive Step: For the inductive step, you assume that $a_k$ satisfies the condition of being either greater than or less than $a_{k+1}$ (depending on whether the sequence is increasing or decreasing).

3. Then, you need to show that $a_{k+1}$ continues to satisfy this condition for $a_{k+2}$.

Analyzing the choices:

• Option (A): Assuming $a_k + 1 \leq a_k$ and showing that $\frac{1}{6 - a_{k+1}} \leq \frac{1}{6 - a_k}$ is useful if you are assuming the sequence is decreasing.

• Option (B): Assuming $\frac{1}{6 - a_k} \leq \frac{1}{6 - a_{k+1}}$ and showing that $a_k \leq a_{k+1}$ is useful if you are assuming the sequence is increasing.

• Option (C): This would assume the sequence is decreasing but reverses the inequality for $a_k$ and $a_{k+1}$. It seems like the wrong direction for a correct inductive step.

Thus, Option (B) is the best choice, because if you assume $\frac{1}{6 - a_k} \leq \frac{1}{6 - a_{k+1}}$, it directly leads to proving the sequence is increasing.

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Final Answer:

(B) Assume $\frac{1}{6 - a_k} \leq \frac{1}{6 - a_{k+1}}$ and show that $a_k \leq a_{k+1}$.

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Would you like a detailed explanation of the full induction proof or more clarification on the steps?

Follow-up questions:

1. How do you prove the base case in induction for sequences?
2. What does it mean for a sequence to be monotonic?
3. How can you generalize induction for different types of sequence relations?
4. What are some common errors when applying mathematical induction to sequences?
5. How can the recursive formula $a_{n+1} = \frac{1}{6 - a_n}$ be analyzed for convergence?

Tip: When working with recursive sequences, try analyzing the behavior of the recursive formula by considering special cases (like $a_1$, $a_2$, etc.) to identify patterns or trends early in the proof.