Question

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

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

${1^2} = 1 \\ {11^2} = 121 \\ {111^2} = 12321 \\ {1111^2} = 1234321 \\ {11111^2} = 123454321 \\ {111111^2} = 12345654321 \\$
First we discuss about that the one square is one, eleven square is $\;121$, one hundred eleven square is $12321$,
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 $1s$ number.
Hence ${111111^2} = 12345654321$.