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The first term in sequence Q equals 1, and for all positive

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by BTGmoderatorDC » Wed Jan 01, 2020 10:14 pm
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




OA D

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by GMATGuruNY » Wed Jan 01, 2020 10:45 pm
BTGmoderatorDC 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
SUM = 1 + |2³-1³| + |3³-2³| + |4³-3³| + |5³-4³| + |6³-5³| + |7³-6³|
All of the red values CANCEL OUT, leaving only the blue value:
SUM = 7³ = 343

The correct answer is D.
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by Scott@TargetTestPrep » Sat Jan 04, 2020 7:20 pm
BTGmoderatorDC 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

OA D

Source: Manhattan Prep

We see that the nth term of sequence Q, when n ≥ 2, is:

a_n = n^3 - (n - 1)^3

Writing the second to seventh terms, we have:

a_2 = 8 - 1, a_3 = 27 - 8, a_4 = 64 - 27, a_5 = 125 - 64, a_6 = 216 - 125 and a_7 = 343 - 216.

Since a_1 = 1, the sum of the first seven terms is:

1 + (8 - 1) + (27 - 8) + (64 - 27) + (125 - 64) + (125 - 64) + (216 - 125) + (343 - 216) = 343

Answer: D

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