Question

Observe the following pattern and supply the missing number.
$
  {11^2} = 121 \\
  {101^2} = 10201 \\
  {10101^2} = 102030201 \\
  {1010101^2} = \ldots \\
  { \ldots ^2} = 10203040504030201 \\
 $

Answer 1

Hint: the analysis of this requires a thinking part. In the given pattern always start with 1 followed by 0 and go up to as many numbers as there are numbers of 1’s given in the number whose square is to be done and then follow the pattern in reverse order.

Complete step-by-step answer:
${101^2} = 10201$
The number of 1’s in $101$ is 2.
So write the number that starts with 1 up to 2 and then decrease by 1 until 1 is reached. The number is of the form
$1 \ldots 2 \ldots 1$
Add zeros in between the dotted lines. The number becomes as
$10201$
${10101^2} = 102030201$
The number of 1’s in $10101$ is 3.
So, write the number that starts with 1 up to 3 and then decrease by 1 until 1 is reached. The number is of the form,
$1 \ldots 2 \ldots 3 \ldots 2 \ldots 1$
Now, add the zeros at the place of the dotted lines. The number becomes as,
$102030201$
Now, after understanding the pattern, let’s calculate the missing number without actual squaring.
${1010101^2} = \ldots $
The number of 1’s in $1010101$ is 4.
So, write the number that starts with 1 up to 4 and then decrease by 1 until 1 is reached. The number is of the form,
$1 \ldots 2 \ldots 3 \ldots 4 \ldots 3 \ldots 2 \ldots 1$
Now, add the zeros at the place of the dotted lines. The number becomes as,
$1020304030201$
Hence, the missing number is $1020304030201$which a square is of $1010101$

${ \ldots ^2} = 10203040504030201$
Now, the maximum number in the squared term is 5.
So , the number of 1 at the alternate places starting with 1 is 5.
So, write the number that starts with 1 and add 1 at the alternate places until 5 one’s are there.
$1 \ldots 1 \ldots 1 \ldots 1 \ldots 1$
Now, add the zeros at the place of the dotted lines. The number becomes as,
$101010101$
Hence, the missing number is $101010101$ whose square give $10203040504030201$


Note: the important point is to understand in squared terms as well as in the number whose square is done, 0 occupies the alternate place.
The number which is obtained by squaring , always starts with 1 followed by 0 then increases the number 1 up to the number of 1's that are present in the squaring term and the process is repeated in reverse order.