Write the next three steps for the given pattern.

$\begin{array}{l}
1 \times 1=1 \\
11 \times 11=121 \\
111 \times 111=12321 \\
1111 \times 1111=1234321 \\
11111 \times 11111=123454321
\end{array}$

Observe the pattern: multiplying a number of n ones by itself results in a palindrome that increases to n and decreases back to 1.
Apply the pattern to find the next three steps.
111111 × 111111 = 12345654321
1111111 × 1111111 = 1234567654321
11111111 × 11111111 = 123456787654321
The next three steps are 12345654321 , 1234567654321 , 123456787654321 ​ .

Explanation

Problem Analysis We are given a pattern of numbers consisting of only the digit 1 multiplied by themselves. We need to find the next three steps in the pattern.

Pattern Recognition The pattern follows that when a number consisting of n ones is multiplied by itself, the result is a palindrome that increases from 1 to n and then decreases back to 1.

Extending the Pattern Following this pattern, the next three steps are:

111111 × 111111 = 12345654321

1111111 × 1111111 = 1234567654321

11111111 × 11111111 = 123456787654321


Examples
This pattern can be seen in various areas, such as computer science when dealing with binary numbers or in signal processing when analyzing symmetrical signals. For example, if you are designing a symmetrical digital filter, understanding patterns like this can help optimize the filter's performance. The palindrome structure ensures that the filter processes signals equally in both directions, which is useful in applications like audio processing or image enhancement.

Answered by GinnyAnswer | 2025-07-08