Answer: Choice C
[tex]t_1 = \frac{4}{243}, \ r = 3[/tex]
==============================================
Work Shown:
Start with [tex]t_6 = 4[/tex] and multiply by the common ratio r to get the next term [tex]t_7[/tex]. Keep this going until you get to the 10th term [tex]t_{10}[/tex]
So we have...
[tex]t_6 = 4\\\\t_7= r*t_6 = 4r\\\\t_8 = r*t_7 = r*(r*t_6) = r^2*t_6 = 4r^2\\\\t_9 = r*t_8 = r*(r^2*t_6) = r^3*t_6 = 4r^3\\\\t_{10} = r*t_9 = r*(r^3*t_6) = r^4*t_6 = 4r^4 = 324\\[/tex]
From that we can see,
[tex]4r^4 = 324\\\\r^4 = \frac{324}{4}\\\\r^4 = 81\\\\r = 81^{1/4}\\\\r = 3[/tex]
At this point, the only possible answer is choice C since this is the only answer choice with r = 3 in it. We could stop here if we wanted.
For the sake of completeness, let's find [tex]t_1[/tex]
[tex]t_n = t_1*(r)^{n-1}\\\\t_{6} = t_1*(3)^{6-1}\\\\4 = t_1*(3)^{5}\\\\4 = t_1*243\\\\t_1 = \frac{4}{243}[/tex]
You can use [tex]t_{10} = 324[/tex] in place of [tex]t_6 = 4[/tex] in the section above, so either value is optional.
