Question

Using the formula: a1/1-r (sum of an infinite geometric series) find the decimal: .7852852852852852... in fraction form.

Answer 1

\[.7852852852852852 .... =\frac{785}{10^3}+\frac{285}{10^6}+\frac{285}{10^9}+\frac{285}{10^{1
2}}=\\
+\frac{785}{10^3}+\frac{285}{10^6}\left(1+\frac{1}{10^3}+\frac{1}{10^6}+\cdots\right)
\]

Answer 2

Use the following. If |r|<1, then\[
1+r + r^2 + r^3 + \cdots = \frac 1{1-r}
\]
To sum the infinite geometric series in my previous post.

Answer 3

You need to sum the infinite series
\[
1+\frac{1}{10^3}+\frac{1}{10^6}+\cdots
\]
Use the above formula in my previuous post with
\[
r=\frac 1{10^3}
\]

Answer 4

\[.7852852852852852 .... =\frac{785}{10^3}+\frac{285}{10^6}+\frac{285}{10^9}+\frac{285}{10^{1
2}}+\cdots=\\
+\frac{785}{10^3}+\frac{285}{10^6}\left(1+\frac{1}{10^3}+\frac{1}{10^6}+\cdots\right)
\]

Answer 5

\[
1+\frac{1}{10^3}+\frac{1}{10^6}+\cdots = \frac {1}{ 1-\frac 1{10^3}} =\frac{1000}{999}
\]

Answer 6

Finally
\[\frac{785}{10^3}+\frac{285}{10^6} \frac{1000}{999}

\]
Simplify and you find your answer.

Answer 7

It seems that you are not online\[
\frac{785}{10^3}+\frac{285}{10^6} \frac{1000}{999}=\frac{523}{666}
\]

Answer 8

In general any repeating decimals can be written as a fraction using the above method.