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The infinite geometric series

x + \frac{1}{2} x^2 + \frac{1}{4} x^3 + \frac{1}{8} x^4 + ...

is equal to 12.  Find the sum of all possible real values of x.

 Aug 11, 2023
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This is an infinite Geometric Sequence with a common ratio of:

 

(x^2/2) / x = x / 2 - This is the common ratio

 

Sum = First term / [1 - comm. ratio]

 

12 = x / [1 - x / 2], solve for x

12 - 6x =x

12 = 7x

x = 12 / 7 - this is value of x as well as the First Term

 

The common ratio = x / 2 = (12/7) / 2 = 6/7

 

Check: [12/7] / [1  -  6/7] =[12/7] / [1/7] = 12

 Aug 11, 2023

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