Question

Use the infinite geometric series formula to write this repeated decimal as a fraction. The decimal is 0.31313131...

Answer 1

The Infinite geometric series formula is:

`\begin{gathered} S_(\infty)=(a)/(1-r) \\ \text{where a is the first term} \\ r\text{ is the common ratio} \end{gathered}`

0.31313131..can be written in a series form as:

`0.31+0.0031+0.000031+0.00000031+\text{.}\ldots`

From this series,

`\begin{gathered} a=0.31=(31)/(100) \\ r=(0.0031)/(0.31)=(1)/(100) \end{gathered}`

Substituting these parameters into the infinite geometric series formula:

We will have:

`\begin{gathered} ((31)/(100))/(1-(1)/(100)) \\ ((31)/(100))/((99)/(100)) \\ (31)/(100)*(100)/(99) \\ (31)/(99) \end{gathered}`

The final answer is 31/99