Answer

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**Hint:**There are 7 ones in \[\left( 1111111 \right)\] . Use the square of 11 and 111, and observe the pattern. The pattern is that for an integer having \[n\] ones, we can write its square simply by writing from 1 to \[n\] and then from \[\left( n-1 \right)\] to 1. Now, solve it further and get the square of \[\left( 1111111 \right)\].

**Complete step-by-step solution**

According to the question, we are given a number and we have to find its square.

The given number = \[\left( 1111111 \right)\] …………………………………………..(1)

We know that,

\[{{\left( 11 \right)}^{2}}=121\] …………………………………….(2)

\[{{\left( 111 \right)}^{2}}=12321\] ……………………………………………..(3)

In equation (2), we can observe that we have to find the square of 11. Here, a pattern can be observed. 111 has two ones. So, counting starts from 1 to 2 and then in decreasing order to 1, i.e., 121.

Similarly, in equation (3), we can observe that we have to find the square of 111. Here, a pattern can be observed. 111 has three ones. So, counting starts from 1 to 3 and then in decreasing order to 1, i.e., 12321.

Therefore, we can think of a pattern here. It can be observed that if we have an integer with \[n\] ones, we can write its square simply by writing from 1 to n and then from n-1 to 1 …………………………………………………(4)

We are asked to find the square of \[{{\left( 1111111 \right)}^{2}}\] .

The number of 1s in the above number = 7 ………………………………………..(5)

From equation (4) and equation (5), we get

The square of 1111111 = 1234567654321.

**Hence, using the pattern, the square of 1111111 is 1234567654321.**

**Note:**Here, someone can try to find the square by multiplying the number. But this process might take a long time and may result in calculation mistakes too. For this type of question, we have to find the square of a number in which all the digits are equal to 1. Use the trick that an integer having \[n\] ones, we can write its square simply by writing from 1 to \[n\] and then from \[\left( n-1 \right)\] to 1.

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