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 is the sum of the first 2 terms of some geometric progression,
while 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 = = = 4
and then r = +/- = +/- 2.
Answer. The common ratio r may have two values: 2 or -2.
Solved.
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Notice. It is possible to make one step further and to determine the first term "a" of the progression.
But since the problem does not ask to do it, I stop at this point.
