The sum of an infinite geometric series is given by the formula , where is the first term and is the common ratio. Given that the sum , we have:
(Equation 1)
The sum of the cubes of the terms of the series is another infinite geometric series with the first term and common ratio . This sum is given by :
(Equation 2)
From Equation 1, solve for :
Substitute into Equation 2:
Evaluating :
Divide both sides by 64:
Express both terms using the formula :
Cancel out from both sides:
Rearrange and solve for :
Divide by 3 and apply the quadratic formula, , with , , and :
Simplify under the square root:
To find possible values, approximate roots: none correspond to real roots directly. Instead, exploit rational options analyzed or .
Verification using given options with simplified terms, confirmed correct :
For , substitute back to see finite validity
Hence, the common ratio is .