Question

I was helping one member today and I got the impression that he/she did not
know the method of writing a repeated decimal in terms of the sum of a geometric series. Here is an explanation of that topics.

Answer 1

I will denote a repeated fraction as
.baaaaaaaa ... ..
For example if b=345 and a =6758 then the repeated fraction will be
.3456758675867586758....
Let n=the length of b, in the example above n=3.
Let m= the length of a, in the example above m=4.
Let us try to write .baaaaaaa... as a sum in terms of a geometirc series.



\[
.baaaaaaaa =\frac b {10^n}+\frac a {10^{m+n}}+\frac a {10^{m^2+n}}+
\frac a {10^{m^4+n}}+ \cdots =\\
\frac b {10^n}+\frac a {10^{m+n}}\left(1+\frac 1 {10^{m}}+
\frac 1 {10^{m^2}}+ \cdots \right)=\\

\frac b {10^n}+\frac a {10^{m+n}}\left(
\frac {1 }
{1- \frac 1 {10^{m}}}
\right)\\
\]

Answer 2

See also
http://en.wikipedia.org/wiki/Geometric_series#Repeating_decimals

Answer 3

Oo ..
this is quite unintuitive