Question

Find the number whose decimal expansion is 0.214214214.... (that is $\overline{0.214}$)

Answer 1

If r is the given decimal number, then we have two relations: $r=\mathrm{0.214214214...}$ and $1000r=\mathrm{214.214214214..}$. Hence $999r=1000r-r=214$. Solving for r, we obtain $r=\frac{214}{999}$