Question

Write the square, making use of a pattern ${\left( {111111} \right)^2}$

Answer 1

Hint:First we construct a pattern and then we discuss that.
We can write the square of \[1`s\] and then find the square of this number.

Complete step-by-step answer:
Now we can construct and then solve by it, using this pattern method
$
  {1^2} = 1 \\
  {11^2} = 121 \\
  {111^2} = 12321 \\
  {1111^2} = 1234321 \\
  {11111^2} = 123454321 \\
  {111111^2} = 12345654321 \\
 $
First we discuss about that the one square is one, eleven square is \[\;121\], one hundred eleven square is \[12321\],
Here we notice that it follows one pattern,
Suppose we take ${11^2}$ as equal to \[121\] , the answer should start with one and end with one.
We can add \[\left( {1 + 1 = 2} \right)\] that is in the middle position.
 Also we can take ${111^2}$ is equal to \[12321\], answer should start and end with one and the middle term is \[\left( {1 + 1 + 1} \right)\] that is equal to \[3\] , and from second position of left and right is \[\left( {1 + 1 = 2} \right)\].
Also we can take ${1111^2}$ is equal to \[1234321\] , answer should start and end with one and the middle term is \[\;\left( {1 + 1 + 1 + 1} \right)\] that is equal to \[4\], and from third position of left and right this is of the form \[\left( {1 + 1 + 1 = 3} \right)\] and from second portion of left and right this is of the form \[\left( {1 + 1 = 2} \right)\]
Also we can take ${11111^2}$ is equal to \[\;123454321\], answer should start and end with one and the middle term is \[\left( {1 + 1 + 1 + 1 + 1} \right)\] that is equal to\[5\], and from fourth position of left and right this is of the form \[\left( {1 + 1 + 1 + 1 = 4} \right)\] and from third position of left and right this is of the form \[\left( {1 + 1 + 1 = 3} \right)\] and from second portion of left and right this is of the form \[\left( {1 + 1 = 2} \right)\]
Similarly we can find up to any \[1`s\] number.
Hence ${111111^2} = 12345654321$.

Note:Here we can use many methods; in general, we can multiply it. In a short time it is best to decode it and using pattern method develop your analyse skills to deeper understand of making hidden patterns inside in it.