Chapter 1.5, Problem 18HP

To determine

Find the value of $1^2,{11}^2,\text{and }{111}^2$ then use your result to predict the value of $111,{111}^2$

Answer to Problem 18HP

  $\begin{array}{l}{1}^2=1\\ {11}^2=121\\ {111}^2=12321\\ 111,{111}^2=12345654321\end{array}$

Explanation of Solution

In order to find the value of $1^2,{11}^2,\text{and }{111}^2$ then use your result to predict the value of $111,{111}^2$ , the number ones determine the solution $11\times 11=121$ , so we have 2 one’s in 11, we stop at 2 and then reverse->1 2 1.

For $111\times 111=12321$ , so we have 3 one’s in 111, we stop at 3 and then reverse->1 2 3 2 1.

Similarly, for $111,{111}^2$ , $111,111\times 111,111=12345654321$ , so we have 6 one’s in 111111, we stop at 6 and then reverse->12345654321.

So, the value of $1^2,{11}^2,\text{and }{111}^2$ and $111,{111}^2$ is 1, 121, 12321, and 12345654321.