Q.Express the recurring decimal $0.\overline{6}$ as a fraction using the sum of an infinite G.P.

[!TLDR] Write $0.\overline6=\dfrac{6}{10}+\dfrac{6}{100}+\cdots$, a G.P. with $a=\tfrac6{10},r=\tfrac1{10}$. [!ANSWER] $0.\overline6=\dfrac23$.

$0.\overline6=0.6666\ldots=\dfrac{6}{10}+\dfrac{6}{100}+\dfrac{6}{1000}+\cdots$, a G.P. with first term $a=\dfrac{6}{10}$ and ratio $r=\dfrac1{10}$. $S_\infty=\dfrac{a}{1-r}=\dfrac{6/10}{9/10}=\dfrac{6}{9}=\dfrac23$. (Check: $2\div3=0.6666\ldots$ ✓.) [!ANSWER] $0.\overline6=\dfrac23$.

Express the recurring decimal as an infinite G.P. with $a=\frac{d}{10}$ and $r=\frac1{10}$, then apply $S_\infty=\frac{a}{1-r}$; the single-digit-repeat shortcut is $d/9$.

Writing the fraction as $\frac{6}{10}$ or forgetting to reduce $\frac69$ to lowest terms $\frac23$ are the common slips.