What does $\frac{1}{9801}$ equal to? While playing on my calculator I accidentally calculated $\frac{1}{9801}$ and the value was $0.00010203040506070809101112131415161718192021222324252627282930\ldots$
We observe that there is a nice pattern in the decimal expansion.
but how long will it continue ie. after how many digits will it repeat?
There is a numberphile video related to this here.

Essentially notice that $9801=99^2$. A similar pattern happens with $\frac{1}{9^2}=\frac{1}{81}$. It's value is $0.\dot{0}1234567\dot{9}$, cycling through all the units except $8$ (To see why, try a long division method). The case with $\frac{1}{99^2}$, it cycles through every two digit number, but excludes $98$ from this list... i.e. it is:
$$\frac{1}{99^2}=0.\dot{0}0010203040506\cdots9596979\dot{9}$$ and similar things happen with $\frac{1}{999^2},\frac{1}{9999^2}$, etc.
Every fraction $\frac{a}{b}$ with integers $a$ and $b$ has a pattern which continues endlessly.