The Quest for 700: Weekly GMAT Challenge (Answer)

Yesterday, Manhattan GMAT posted a 700 level GMAT question on our blog. Today, they have followed up with the answer:

The best way to approach this problem is to generate an actual table according to the rules. Let’s start with the most concrete information we know: the entry in the upper left corner is 1. Now let’s build the table to the right and then down.

According to the second rule, entries double as we go to the right, so the first row looks like this:
1   2   4   8   16 …

According to the first rule, entries “flipflop” in sign (+/–) as we go down the table. So the first five rows and columns look like this:
1   2   4   8   16
-1  -2  -4  -8  -16
1   2   4   8   16
-1  -2  -4  -8  -16
1   2   4   8   16

Since n is an odd integer greater than 4, we can choose n = 5 and stop here.

The process of adding up all the entries benefits from cancelation. The first four rows cancel each other out completely. All that is left is the last row, which sums up as follows: 1 + 2 + 4 + 8 + 16 = 31. Plugging n = 5 into the answer choices, we get the following:

(A) 0    INCORRECT
(B) n² – 1 = 5² – 1 = 24    INCORRECT
(C) n² + 1 = 5² + 1 = 26    INCORRECT
(D) 2ⁿ – 1 = 2⁵ – 1 = 31     CORRECT
(E) 2ⁿ + 1 = 2⁵ + 1 = 33    INCORRECT

We can prove that 2ⁿ – 1 is the general formula for the sum of 1, 2, 4, etc. up to 2^{n – 1} (which will be the last entry in the last row), but there is no need to do so.

The correct answer is (D).