SOLUTION: The first four terms of gp is 30 and that of the last four term is 960 if the first term of gp is 1 find common ratio

If the first term is 1, then from the first part of the condition we have

 = 30.    (1)

I don't know how to solve it analytically for r  (if do not use the Cardano's formula, which is, obviously, not from this opera).


See the plot





Plot y = 


I created a table in Excel, which shows that r is between 2.67 and 2.68,
but it is clear that the graphical method (or any approximate "technology" method) 
is not what is expected as the solution method for this problem.


The second condition does not give much:

it gives the second equation

 = 960,    (2)

with some unknown "n", which combined with (1) gives  =  = 32.

but the condition does not give any clue for the value of n.

So, although the "technology" can provide some approximate solution, 
looking into the problem "in large", I don't think that the condition is correct.


Again:

the problem (as it is presented in the post) is reduced to two equations in two unknowns r and n

 = 30,     (1)    and
 = 32,             (2)

and I don't want (and don't know how) to go further.


It doesn't look to be attractive.