Given
The formula for the sum of an infinite geometric series
![s =\frac{a1}{1-r}](https://tex.z-dn.net/?f=s%20%3D%5Cfrac%7Ba1%7D%7B1-r%7D)
used to convert 0.23 to a fraction
Find out the values of a1 and r
To proof
As given in the question
The series is infinte geometric series
The geometric is mostly in the form a , ar , ar² ,............
s = a + ar + ar² + ar³...........
Where a = first term
r = common ratio
The series is infinte geometric series i.e 0.2323...
thus it is written in the form
s = 0.23 + 0.0023 + 0.000023 +...........
now written in the fraction form
we get
![s = \frac{23}{100} +\frac{23}{10000}+\frac{23}{1000000} + .......](https://tex.z-dn.net/?f=s%20%3D%20%5Cfrac%7B23%7D%7B100%7D%20%2B%5Cfrac%7B23%7D%7B10000%7D%2B%5Cfrac%7B23%7D%7B1000000%7D%20%2B%20.......)
compare this series with the
s = a + ar + ar² + ar³...........
Thus
we get
![a1 = \frac{23}{100}](https://tex.z-dn.net/?f=a1%20%3D%20%5Cfrac%7B23%7D%7B100%7D)
a1 = 0.23
![r = \frac{1}{100}](https://tex.z-dn.net/?f=r%20%3D%20%5Cfrac%7B1%7D%7B100%7D)
r = 0.01
Hence proved