4. Describe an algorithm that inserts an integer x in the appropriate position into the list a1,a2, . . . , an of integers that are in increasing order

1. Initialize: Start with a list of integers $a_1, a_2, \ldots, a_n$ that are already sorted in increasing order. Let $x$ be the integer you want to insert.

2. Find the Position: Iterate through the list to find the appropriate position for $x$. This can be done using a loop:

   • Initialize an index variable $i$ to 0.
   • While $i < n$ and $a_i < x$, increment $i$ by 1.

3. Insert the Integer: Once the correct position $i$ is found:

   • Shift all elements from position $i$ to $n-1$ one position to the right to make space for $x$.
   • Insert $x$ at position $i$.

4. Update the List: The list now contains $n+1$ elements, with $x$ inserted in the correct position to maintain the increasing order.

Here is a step-by-step example:

• Given list: $a = [1, 3, 5, 7]$
• Integer to insert: $x = 4$

Step-by-Step Execution:

1. Initialize $i = 0$.
2. Compare $x$ with $a[i]$:
   • $x = 4$, $a[0] = 1$ (since $4 > 1$, increment $i$).
   • $x = 4$, $a[1] = 3$ (since $4 > 3$, increment $i$).
   • $x = 4$, $a[2] = 5$ (since $4 < 5$, stop incrementing $i$).
3. Insert $x$ at position $i = 2$:
   • Shift elements from position 2 to the right: $a = [1, 3, 5, 5, 7]$.
   • Insert $x$ at position 2: $a = [1, 3, 4, 5, 7]$.

The final list is $[1, 3, 4, 5, 7]$.