Question:
The sum of an infinite geometric series is $27$ times the series that results if the first three terms of the original series are removed. What is the value of the series' common ratio?

Answer:
Let's denote the first term as $a$ and the common ratio as $r.$ Additionally, call the original sum of the series $S.$ It follows that \[\frac{a}{1-r}=S.\] After the first three terms of the sequence are removed, the new leading term is $ar^3.$ Then one $27^{\text{th}}$ of the original series is equivalent to \[\frac{ar^3}{1-r}=r^3\left( \frac{a}{1-r}\right)=\frac{S}{27}.\]

Dividing the second equation by the first, $r^3= \frac{1}{27}$ and $r=\boxed{\frac{1}{3}}.$