How do you Use an infinite geometric series to express a repeating decimal as a fraction?

Question

Evelyn Agard · Accepted Answer

Let us find a fraction for

#0.121212...#

by splitting it into

#=0.12+0.0012+0.000012+cdots#

by rewriting each term as a fraction,

#=12/100+12/10000+12/1000000+cdots#

by rewriting a little further,

#=12/100+12/100(1/100)+12/100(1/100)^2+cdots#

The series above is a geometric series with

#a=12/100# and #r=1/100#

Hence, the sum is

#0.121212...=a/{1-r}={12/100}/{1-1/100}={12/100}/{99/100}=12/99=4/33#

Isabella Ackerman · Answer

undefined To express a repeating decimal as a fraction using an infinite geometric series, follow these steps:

1. Identify the repeating pattern in the decimal.
2. Write the repeating pattern as a fraction without the repeating part over the appropriate number of 9s (for example, if the repeating pattern is 0.363636..., write it as 36/99).
3. Express the fraction in its simplest form.
4. Recognize that the repeating decimal can be expressed as a sum of an infinite geometric series with the first term being the non-repeating part of the decimal and the common ratio being 0.1 raised to the power of the number of decimal places in the repeating part.
5. Use the formula for the sum of an infinite geometric series to find the fraction equivalent to the repeating decimal.

Here's the formula for the sum of an infinite geometric series:

$$ S = \frac{a}{1 - r} $$

Where:
- $ S $ is the sum of the series.
- $ a $ is the first term of the series.
- $ r $ is the common ratio of the series (in this case, it's 0.1 raised to the power of the number of decimal places in the repeating part).

Plug in the values of $ a $ and $ r $ into the formula and simplify to find the fraction equivalent to the repeating decimal.