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.
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
