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# Find the square of ${{\left( 1111111 \right)}^{2}}$.

Last updated date: 11th Aug 2024
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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.