Question

The integer closest to $\sqrt {111.....1 - 222.....2} $, where there are $2018$ ones and $1009$ twos, is
A. $\dfrac{{{{10}^{1009}} - 1}}{3}$
B. $\dfrac{{{{10}^{1009}} - 1}}{9}$
C. $\dfrac{{{{10}^{2018}} - 1}}{3}$
D. $\dfrac{{{{10}^{2018}} - 1}}{9}$

Answer 1

Hint: We have to first find the value of the expression using $2$ ones and $1$ two. Then we have to increase the number of ones and twos and find how they can be represented as a general formula. We need to find a pattern that matches the expression. Here the number of ones is double the number of twos. We have to manipulate the value in the given format.

Complete answer:
It is given to find the integer that is closest to the expression $\sqrt {111.....1 - 222.....2} $ consisting of $2018$ ones and $1009$ twos.
We have to start with a small number of ones and twos to get a pattern in the value of the expression.
So we first calculate the value of expression having $2$ ones and $1$ two.
$\therefore $ The required value is
$
\sqrt {11 - 2} \\
= \sqrt 9 \\
= 3 \\
= \dfrac{9}{3} \\
= \dfrac{{{{10}^1} - 1}}{3} \\
$
Similarly, we calculate the value of expression having $4$ ones and $2$ twos.
$\therefore $ The required value is
$
\sqrt {1111 - 22} \\
= \sqrt {1089} \\
= 33 \\
= \dfrac{{99}}{3} \\
= \dfrac{{{{10}^2} - 1}}{3} \\
$
Again, we calculate the value of expression having $6$ ones and $3$ twos.
$\therefore $ The required value is
$
\sqrt {111111 - 222} \\
= \sqrt {110889} \\
= 333 \\
= \dfrac{{999}}{3} \\
= \dfrac{{{{10}^3} - 1}}{3} \\
$
To calculate the expression $\sqrt {111.....1 - 222.....2} $, where there are $2018$ ones and $1009$ twos, is
$
\sqrt {111.....1 - 222.....2} \\
= 333....3 \\
= \dfrac{{999....9}}{3} \\
= \dfrac{{{{10}^{1009}} - 1}}{3} \\
$
The simplified value of this expression is $333....3$ ($1009$ times).
Hence, the correct option is (a) $\dfrac{{{{10}^{1009}} - 1}}{3}$.

Therefore, the correct option is A

Note: If the number of ones and twos were the same then this formula will not be applied. The number of digits in the result of the expression is the same as the number of twos in the expression. We have to manipulate the result so that it matches with any one of the options. The expression cannot be calculated manually so we have to find a pattern to calculate this.