As Brent says above, you don't need to know anything about "geometric series" for the GMAT - you don't even need to know what they are. So ignoring the other problems with the wording of the question, it's simply out of the scope of the test. The only way it could appear is if the question itself also provided the formulas you'd need to use to answer it.
But since the only solution above assumes r is a positive integer, and uses inspection, I can offer a different method, though test takers can safely ignore questions like this. We know from Statement 1 that the first term a = 1, and the last term is 1024. In a geometric sequence with n terms, the last term is equal to (a)(r^(n-1)). Since a =1, we know
r^(n-1) = 1024
The sum S of a geometric sequence with n terms is equal to
S = a(1- r^n)/(1 - r)
so since that sum is 2047, we also know, using that a=1,
2047 = (1 - r^n) / (1 - r)
Since r^n is just equal to (r)(r^(n-1)) by basic exponent rules, and since r^(n-1) = 1024, we can replace r^n with 1024r. So
2047 = (1 - 1024r) / (1 - r)
2047 - 2047r = 1 - 1024r
2046 = 1023r
2 = r
and Statement 1 is sufficient. Statement 2 is not sufficient (it could be the sequence with r=2 that we found in Statement 1, but since r need not be an integer, it could just be a two-term sequence where the first term is 512, and the second is 1535, among many possibilities), so the answer is A.
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