Geometric Series problem

**bobsanchez** · August 23rd 2011, 04:06 PM

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.

**Plato** · August 23rd 2011, 04:18 PM

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?

**skeeter** · August 23rd 2011, 04:34 PM

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

**bobsanchez** · August 23rd 2011, 05:25 PM

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.

**skeeter** · August 23rd 2011, 05:31 PM

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

$\frac{12}{99}$

**bobsanchez** · August 24th 2011, 05:44 AM

Okay, so a = 12/10?

**skeeter** · August 24th 2011, 02:00 PM

Originally Posted by bobsanchez

Okay, so a = 12/10?

the first term of the geometric series is $\frac{12}{100}$

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