Question

The digit at the ${100}^{th}$ place in the decimal representation of $\frac{6}{7}$ is

• A. 1
• B. 2
• C. 4
• D. 5

Answer 1

The correct option is A 1

$\frac{6}{7}=0.857142\mathrm{85714..}$
Hence, the number is repeating itself. Now, let the number occurring at 100th place be he same as that occurring at $x^{th}$ place from the first 6 digits. And each number repeats itself after 6 digits. Since, $6\ast 16=96$, the next will be 17th set of the numbers repeating itself.
Hence, by applying AP,
$a_n=a_1+(n-1)d$
$100=x+(17-1)\left(6\right)$
$100=x+96$
$x=4$
Thus, the number at 4th place will be same as that at 100th place i.e. 1