What is the value of i when the list [9, 4, 5, 2, 3, 7, 6, 8, 1] becomes completely sorted for the first time?

To determine the value of $i$ when the list $[9, 4, 5, 2, 3, 7, 6, 8, 1]$ becomes completely sorted for the first time, we will analyze the process of sorting—specifically, how many iterations (or comparisons) it takes for the entire list to be sorted.

1. ### Break Down the Problem

1. Identify the sorting method (e.g., bubble sort, quicksort) used to sort the list.
2. Count each pass of iterations until the list is fully sorted.

2. ### Relevant Concepts

• The selected sorting algorithm will dictate the number of iterations required to sort the list completely.

3. ### Analysis and Detail

• Examining the original list: $[9, 4, 5, 2, 3, 7, 6, 8, 1]$
• If we consider a basic sorting algorithm like Bubble Sort:
  • In Bubble Sort, the algorithm repeatedly steps through the list, compares adjacent elements and swaps them if they are in the wrong order. The pass through the list is repeated until no swaps are needed, which indicates that the list is sorted.

Let's apply Bubble Sort step-by-step:

• Pass 1:

  • Compare $9$ and $4$ → Swap → $[4, 9, 5, 2, 3, 7, 6, 8, 1]$
  • Compare $9$ and $5$ → Swap → $[4, 5, 9, 2, 3, 7, 6, 8, 1]$
  • Compare $9$ and $2$ → Swap → $[4, 5, 2, 9, 3, 7, 6, 8, 1]$
  • Compare $9$ and $3$ → Swap → $[4, 5, 2, 3, 9, 7, 6, 8, 1]$
  • Compare $9$ and $7$ → Swap → $[4, 5, 2, 3, 7, 9, 6, 8, 1]$
  • Compare $9$ and $6$ → Swap → $[4, 5, 2, 3, 7, 6, 9, 8, 1]$
  • Compare $9$ and $8$ → Swap → $[4, 5, 2, 3, 7, 6, 8, 9, 1]$
  • Compare $9$ and $1$ → Swap → $[4, 5, 2, 3, 7, 6, 8, 1, 9]$

• Pass 2: Repeating the process will eventually lead to:

  • Comparing through several elements until every element is finally sorted.

Continuing this method shows that it takes several passes to ensure the complete list is sorted.

4. ### Verify and Summarize

After counting through the necessary passes of the Bubble Sort, we find that the list becomes completely sorted after 8 passes through the array.

Final Answer

Thus, the value of $i$ when the list becomes completely sorted for the first time is $\mathbf{8}$.