Example 2 There is a sequence of numbers, the 1st number is 105, the 2nd number is 85, starting from the 3rd number, each number is the average of the two numbers before it. What is the integer part of the 19th number?
Notice that "the average of any two numbers $a$ and $b$, $\frac{a+b}{2}$, corresponds to the midpoint of the segment between the points representing $a$ and $b$ on the number line," we can consider placing these 19 numbers on the number line, as shown in Figure 2.

From the property mentioned above, starting from the 6th number, the numbers represented on the number line are all between 91 and 92. Therefore, the integer part of the 19th number is 91.