geometric sequence question?

If the TERMS are the "S"s, then

S13 = (S12)^2/S11 = 1016.816

If the SUMS are the "S"s, then

we don't really have enough information;

we know only that the 12th term was 111.

Using brute force methods (and WolframAlpha), I found that a ≈ 45.9235349 and r ≈ 1.083538466. I used the formula Sn = a(r^n - 1)/(r - 1) and got expressions for S11 and S12. I divided S12 by S11 to get

890/779 = (r^12 - 1)/(r^11 - 1)

I expanded that to get a 12th degree polynomial equation in r and asked WolframAlpha for solve for r. The above approximation was the only solution that made sense. Then I plugged this r into S11 and got "a". I plugged these into the equation for S12 and got very nearly 890. With these you can compute a13 and add it to S12 to get S13. I get about 1010.27277. Obviously, this is not the way you are supposed to do this problem anyway. I had hoped that the approximation to S13 would give me a hint as to the "right" way, but it hasn't. Leave this question open another day and I'll see if I can figure out the clever way to do this problem (I am convinced it does not require the determination of a or r).

Edit: No joy. I got to S_(n+1) = r*S_n + a, which enables us (using S11 and S12) to get one of r and a in terms of the other, but I still think this is the wrong approach.