Question

Observe the following pattern and find the missing digits.
$\begin{align}
  & {{11}^{2}}=121 \\
 & {{101}^{2}}=10201 \\
 & {{1001}^{2}}=1002001 \\
 & {{100001}^{2}}=10000200001 \\
 & {{10000001}^{2}}=.......... \\
\end{align}$

Answer 1

Hint: Check each number one by one and see the pattern of the digits 0, 1 and 2 in each of the four expressions given. See that when there is no 0 in the first number so there is no zero in the R.H.S. when there is one 0 in the L.H.S so there is one in the R.H.S before and after 2. For two and four 0’s in the L.H.S we have two and four in the R.H.S respectively and that also before and after 2. Similarly, see the position of digit 2 and accordingly write the answer.

Complete step by step solution:
Here we have been provided with some numbers forming a particular pattern and we are asked to find the missing number. Let us see the pattern carefully.
$\begin{align}
  & {{11}^{2}}=121 \\
 & {{101}^{2}}=10201 \\
 & {{1001}^{2}}=1002001 \\
 & {{100001}^{2}}=10000200001 \\
 & {{10000001}^{2}}=.......... \\
\end{align}$
Let us first see the pattern of the digit 0. We conclude that when there is no 0 in the first number so there is no zero in the R.H.S. when there is one 0 in the L.H.S so there is one in the R.H.S before and after 2. For two and four 0’s in the L.H.S we have two and four in the R.H.S respectively and that also before and after 2. Now, the number present at the bottom has six 0’s that means in the missing place the number will have six 0’s before and after the digit 2. As for the digit 1 we see that they are present at the beginning and at the end. So the missing number can be given as:
$\Rightarrow {{10000001}^{2}}=100000020000001$
Hence the above expression is our answer.

Note: If you want you can check the answer by squaring the number 10000001, however it will be difficult as it is a very large number so the chances of making the calculation mistake will be higher. So it is fine to check the answer by squaring but it will not be a good approach if you will use it to find the answer.