Answer
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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.
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.
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