SOLUTION: Kindly solve this one for me: The sum of all the terms of an infinite geometric series is 4 while the sum of the cubes of all the term is 192. What are the terms?

Question 983221: Kindly solve this one for me:
The sum of all the terms of an infinite geometric series is 4 while the sum of the cubes of all the term is 192. What are the terms?
Answer by ikleyn(52788)   (Show Source): You can put this solution on YOUR website!

Let a be the first term of our geometric progression and r be its common ratio.

Then our first equation is

= .

It is the formula for the sum of infinite geometric progression.

The sequence consisting of cubes of geometric progression is geometric progression itself, with the first term    and the common ratio . It is easy to check.

Therefore, the sum of such progression is

= .

It is our second equation.

Now, divide the second equation by the first one. You will get

= .

Thus you decreased the degree of the second equation from 3 to 2.

Next, express the term a from the first equation as

=

and substitute it into the third equation. You will get

= ,   or

= ,   or

= .

It is a quadratic equation.

Can you solve it and complete the solution from this point?

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