The Quest for 700: Weekly GMAT Challenge (Answer)

To solve this problem, we must translate the verbal instructions for the construction of the sequence. The first term is easy: Q1 = 1. 

Next, we have this difficult wording: “for all positive integers n equal to or greater than 2, the nth term in sequence Q equals the absolute value of the difference between the nth smallest positive perfect cube and the (n-1)st smallest positive perfect cube.”

So we’re dealing with all the positive integers beyond 1. Let’s take as an example n = 2. The instructions become these: “the second term equals the absolute value of the difference between the second (nth) smallest positive perfect cube and the first (that is, n-1st) smallest positive perfect cube.”

We know we need to consider the positive perfect cubes in order:
13 = 1 = smallest positive perfect cube (or “first smallest”).
23 = 8 = second smallest positive perfect cube.

The absolute value of the difference between these cubes is 8 – 1 = 7. Thus Q2 = 8 – 1 = 7.
Likewise, Q3 = |33 – 23| = 27 – 8 = 19, and so on.
Now, rather than figure out each term of Q separately, then add up, we can save time if we notice that the cumulative sums “telescope” in a simple way. This is what telescoping means:

The sum of Q2 and Q1 = 7 + 1 = 8. We can also write (8 – 1) + 1 = 8. Notice how the 1’s cancel.

The sum of Q3, Q2 and Q1 = 19 + 7 + 1 = 27. We can also write (27 – 8) + (8 – 1) + 1 = 27. Notice how the 8’s and the 1’s cancel. 

At this point, we hopefully notice that the cumulative sum of Q1 through Qn is just the nth smallest positive perfect cube.

So the sum of the first seven terms of the sequence is 73, which equals 343.