Chapter 9.3, Problem 41E

Solution

To identify: What is the limit if pair up with a partner to create a sequence: Each of you

picks five random digits from 1 to 9 (with repetitions, if you wish). Merge your

digits to make a list of ten. Now each of you constructs a different ten-digit

number using exactly the digits in your list.

Let $a_1$ = the (positive) difference between your two numbers.

Let $a_{n+1}$ = the sum of the digits of $a_n$

  $n\ge 1.$

This sequence converges because it is eventually constant.

The limit of the sequence is $9$ .

Given information:

The given sequence is $\left\{1,2,3,4,5,6,7,8,9\right\}$

Explanation:

Consider the given sequence.

Two possible sets of five numbers:

Set 1: $\left\{1,2,4,5,8\right\}$

Set2: $\left\{3,4,7,7,9\right\}$

By merging the two sets, we obtain the 10 element set: $\left\{1,2,3,4,4,5,7,7,8,9\right\}$

Two possible ten digit numbers we can create using this set are: $1234457789\text{and}\text{9877544321}$

Using the given recursive rule,

  $a_1=\text{9877544321}-1234457789=8643086532$

Similarly

  $\begin{array}{c}{a}_2=8+6+4+3+0+8+6+5+3+2=45\\ a_3=4+5+9\end{array}$

Thus, the limit of the sequence is $9$ .