Question

The formula for the sum of an infinite geometric series, s=a1/1-r, may be used to convert 0.23 to a fraction. What are the values of a1 and r?

Answer 1

Given

The formula for the sum of an infinite geometric series

$s =\frac{a1}{1-r}$

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

$s = \frac{23}{100} +\frac{23}{10000}+\frac{23}{1000000} + .......$

compare this series with the

s = a + ar + ar² + ar³...........

Thus

we get

$a1 = \frac{23}{100}$

a1 = 0.23

$r = \frac{1}{100}$

r = 0.01

Hence proved

Answer 2

I believe the number to be converted to a fraction is 0.232323..., this is 0.23 periodic.

The fomula is right:

 $S= \frac{ a_{1} }{1-r}$

Where $a_{1}$  is 0.23 and r is 1/100

Check it here:

    23/100          23/100         23
-------------- =    ----------- =   ------- = 0.232323...
1 - 1/100          99/100          99

So, the answer is   $a_{1}=$   23/100 and r = 1/100