Question 1123877
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From the post, I hardly can read the small symbol " t " with subscript " x " and have no idea what does it mean.


But I guess, and my guessing is that  {{{S[2]}}}  is the sum of the first 2 terms of some geometric progression, 
while  {{{S[4]}}}  is the sum  of the first 4 terms of the same geometric progression.


If so, then this info can be presented as these two equations


a + ar = 10,                      (1)

a + ar + ar^2 + ar^3 = 50         (2)


and the problem is to find the common ratio "r" from these equations.


We can rewrite equation (2) in the form

(a+ar) + r^2*(a+ar) = 50.


Replacing  (a+ar)  by 10  in this equation, based on (1), gives

10 + 10r^2 = 50,


which implies  r^2 = {{{(50-10)/10}}} = {{{40/10}}} = 4  


and then  r = +/- {{{sqrt(4)}}} = +/- 2.


<U>Answer</U>.  The common ratio r may have two values: 2 or -2.
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Solved.


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<U>Notice</U>.  &nbsp;&nbsp;It is possible to make one step further and to determine the first term "a" of the progression.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;But since the problem does not ask to do it, &nbsp;I stop at this point.