Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:
Our first task is to ensure that we understand the definition of a geometric sequence. Let’s use the sequence given to us: a, b, c, d. We are told that the ratio of any term after the first to the preceding term is a constant. In other words, b/a = some constant, which is the same constant for the other ratios (c/b and d/c). Let’s name that constant r. Thus, we have the following:
b/a = c/b = d/c = r
By a series of substitutions, we can rewrite the sequence in terms of just a and r:
a, b, c, d is the same as a, ar, ar2, ar3
Rewriting the sequence this way highlights the role of the constant ratio r. That is, to move forward in the sequence one step, we just multiply by a constant factor r. Rewriting also lets us substitute into the alternative sequences and watch what happens.
I. dk, ck, bk, ak
This sequence is the same as ar3k, ar2k, ark, ak. To move forward in the sequence, we divide by r. This is the same thing as multiplying by 1/r. Since this factor is constant throughout the sequence, the sequence is geometric.
II. a + k, b + 2k, c + 3k, d + 4k
This sequence is the same as a + k, ar + 2k, ar2 + 3k, ar3 + 4k. To move forward in this sequence, we cannot simply multiply by a constant expression. The presence of the plus sign means that we will not have a constant ratio between successive terms, and this sequence is not geometric.
III. ak4, bk3, ck2, dk
This sequence is the same as ak4, ark3, ar2k2, ar3k. To move forward in the sequence, we multiply by r and divide by k. In other words, we multiply by r/k, which is a constant factor. This sequence is therefore geometric.
Only sequences I and III are geometric.
The correct answer is (D).