1. ## Geometric Series problem

What is the geometric series that corresponds to the periodic decimal

x = .1212121212 . . . ?

(i) Use the formula for the sum of a geometric series to compute the value of x as a fraction.

(ii) What is the value of the periodic decimal y = .99999 . . . ?

From my notes, I'm not quite sure where to go with this. The examples dealt with .333 and other patterns composed of the same number, and are different in other ways, so I'm not really sure where to start.

Any help is really, really appreciated.

2. ## Re: Geometric Series problem

Originally Posted by bobsanchez
What is the geometric series that corresponds to the periodic decimal x = .1212121212 . . . ?
$\sum\limits_{k = 1}^\infty {\frac{{12}}{{10^{2k} }}}$ WHY?

3. ## Re: Geometric Series problem

Originally Posted by bobsanchez
What is the geometric series that corresponds to the periodic decimal

x = .1212121212 . . . ?

(i) Use the formula for the sum of a geometric series to compute the value of x as a fraction.

(ii) What is the value of the periodic decimal y = .99999 . . . ?

From my notes, I'm not quite sure where to go with this. The examples dealt with .333 and other patterns composed of the same number, and are different in other ways, so I'm not really sure where to start.

Any help is really, really appreciated.
$x = 0.12 + 0.0012 + 0.000012 + ...$

$x = \frac{12}{100} + \frac{12}{10000} + \frac{12}{100000} + ...$

$x = \frac{12}{10^2} + \frac{12}{10^4} + \frac{12}{10^6} + ...$

common ratio, $r = \frac{1}{10^2} < 1$

now use $\frac{a_1}{1-r}$ to determine the sum as a fraction

4. ## Re: Geometric Series problem

I somehow ended back up with .12/.99

That equals .121212... but I'm still not sure if that's what I'm after or not.

5. ## Re: Geometric Series problem

(i) Use the formula for the sum of a geometric series to compute the value of x as a fraction.
$\frac{12}{99}$

6. ## Re: Geometric Series problem

Okay, so a = 12/10?

7. ## Re: Geometric Series problem

Originally Posted by bobsanchez
Okay, so a = 12/10?
the first term of the geometric series is $\frac{12}{100}$