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
$$\begin{gathered} 1^2=1 \\ {11}^2=121 \\ {111}^2=12321 \\ {1111}^2=1234321 \\ {11111}^2=123454321 \\ {111111}^2=12345654321 \end{gathered}$$
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.