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?

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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) About Me  (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

a%2F%281-r%29 = 4.

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  a%5E3  and the common ratio  r%5E3.  It is easy to check.

Therefore,  the sum of such progression is

a%5E3%2F%281-r%5E3%29 = 192.

It is our second equation.

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

a%5E2%2F%281%2Br%2Br%5E2%29 = 48.

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

Next,  express the term  a  from the first equation as

a = 4%2A%281-r%29

and substitute it into the third equation.  You will get

%284%5E2%2A%281-r%29%5E2%29%2F%281%2Br%2Br%5E2%29 = 48,   or

%281-r%29%5E2 = 3%2A%281%2Br%2Br%5E2%29,   or

1+-2r+%2B+r%5E2 = 3r%5E2+%2B+3r+%2B3.

It is a quadratic equation.

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