Bunuel wrote:

The first term in sequence Q equals 1, and 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. The sum of the first seven terms in sequence Q is

(A) 91

(B) 127

(C) 216

(D) 343

(E) 784

Kudos for a correct solution.

MANHATTAN GMAT OFFICIAL SOLUTION:To solve this problem, we must translate the verbal instructions for the construction of the sequence. The first term is easy: \(Q_1 = 1\).

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:

1^3 = 1 = smallest positive perfect cube (or “first smallest”).

2^3 = 8 = second smallest positive perfect cube.

The absolute value of the difference between these cubes is 8 – 1 = 7. Thus \(Q_2 = 8 – 1 = 7\).

Likewise, \(Q_3 = |3^3 – 2^3| = 27 – 8 = 19\), 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 \(Q_2\) and \(Q_1 = 7 + 1 = 8\). We can also write (8 – 1) + 1 = 8. Notice how the 1’s cancel.

The sum of \(Q_3\), \(Q_2\) and \(Q_1 = 19 + 7 + 1 = 27\). 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 \(Q_1\) through \(Q_n\) is just the nth smallest positive perfect cube.

So the sum of the first seven terms of the sequence is 7^3, which equals 343.

The correct answer is D.
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