Determining 'r' when given Infinite Sum and Sum of Terms

[Yendall]

This is really similar to the thread I started ealier, however a lot harder to solve.

I was just doing a NEAP practice exam and came across a question very similar to that of the previous thread:

The sum of the first three terms of a geometric series is 57 while the infinite sum is 81.
The common ratio is:

I know that the infinite sum will be: OR

I know that the sum of n terms will be: OR

When entered into CAS like this:


It returns the answer:


Is it okay to conclude that when presented with an answer like this?

From there I could solve for r by:





However, that's the incorrect solution for 'r'. How do I solve something like this?

Thanks for any help.

[Phy124]

I didn't do further but I think I've concluded you are trying to find and , using the equations





In which case you can solve them simultaneously on your calculator:

When entered into CAS like this:


It returns the answer:


Is it okay to conclude that when presented with an answer like this?

That answer isn't telling you anything, it is just a different form of what you inputted, as you are trying to solve one equation for two variables.

edit: had solved using 91 and not 81, fixed now

[Yendall]

I didn't do further but I think I've concluded you are trying to find and , using the equations





In which case you can solve them simultaneously on your calculator:

When entered into CAS like this:


It returns the answer:


Is it okay to conclude that when presented with an answer like this?

That answer isn't telling you anything, it is just a different form of what you inputted, as you are trying to solve one equation for two variables.

edit: had solved using 91 and not 81, fixed now

Thaaaank you!