What is a recursive definition for the sequence a_n when a_n is defined as n times a_{n-1} plus 1?

The recursive definition of the sequence a can be derived by analyzing how each term relates to the previous terms in the sequence. Here’s a detailed explanation:

1. **Base Case**: To begin the recursion, we need to provide a starting point. Let’s define the first term:

• a₁ = 1

This means that the first term in the sequence is 1.

2. **Recursive Step**: The next part of the definition involves creating the subsequent terms based on the previous term. According to the problem, for n > 1, we have:

• a_n = n * a_n-1 + 1

This means that each term a_n is obtained by multiplying the term’s index n by the previous term a_{n-1} and then adding 1.

3. **Summary of the Definition**: Combining both parts together, we have the full recursive definition:

• Base Case: a₁ = 1
• Recursive Step: a_n = n * a_n-1 + 1 for n > 1

Using this recursive definition, we can generate terms in the sequence:

• a₂ = 2 * a₁ + 1 = 2 * 1 + 1 = 3
• a₃ = 3 * a₂ + 1 = 3 * 3 + 1 = 10
• a₄ = 4 * a₃ + 1 = 4 * 10 + 1 = 41

And so forth. Therefore, this recursive definition provides a systematic way to calculate the values of the sequence a.