Q.Find the $n$th term of the sequence $0.6, 0.66, 0.666, 0.6666, \ldots$

Any purely or eventually recurring decimal can be written as an infinite G.P. and therefore converted into an exact fraction using the sum-to-infinity formula. For example, $0.666\ldots=0.6+0.06+0.006+\cdots$ is a G.P. with $a=0.6$ and $r=0.1$, so its sum to infinity is $\dfrac{0.6}{1-0.1}=\dfrac23$. A block of $k$ repeating digits gives common ratio $10^{-k}$; a two-digit repeat such as $0.\overline{46}$ has $a=0.46, r=0.01$, giving $\dfrac{46}{99}$. When a non-recurring part precedes the recurring block, such as $2.\overline5$, the leading part is added separately to th …