Question

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

Answer 1

To write the number as fraction we will first write the number as a geometric series by of the form $a+ar+a{{r}^{2}}+...$ splitting the repeated terms. Now we know that for any GP its infinite sum is given by the formula $\dfrac{a}{1-r}$ . Hence on substituting the values of a and r and then simplifying we will get the fractional representation of the given number.

Complete step-by-step solution:
Now to convert a recurring decimal into fraction we can use the sum of infinite geometric series.
To do so we will first write the recurring decimal as a geometric series.
To do so we will split the number for each repeating part.
For example consider the decimal 0.1111111…..
The number 0.11111… can be written as 0.1 + 0.01 + 0.001 + 0.0001 + ….
Now we can see that the series is a GP and the common ratio in the GP is 0.1.
Hence we have for this GP a = 0.1 and r = 0.1.
Now we know that the sum of infinite GP is given by ${{S}_{\infty }}=\dfrac{a}{1-r}$
Hence we get, ${{S}_{\infty }}=\dfrac{0.1}{1-0.1}=\dfrac{0.1}{0.9}=\dfrac{1}{9}$ .
Hence we have $0.1+0.01+0.001+...=\dfrac{1}{9}$
Hence the fractional representation of 0.11111… is $\dfrac{1}{9}$.

Note: Note that to write the decimal as a GP first term is the number without repetition then count the number of digits that are repeating after decimal. If n is the number of digits then $r=\dfrac{1}{{{10}^{n}}}$ . Hence we can easily write the number in as geometric series. Also note that suppose we have a number of the form 1.2343434… we will first split it into 1.2 + 0.34 + 0.0034 +… and then solve the GP for 0.34 + 0.0034 + …