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09 Sep 2015, 00:58
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
69% (03:03) correct
31%
(02:57)
wrong
based on 596
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Sequence S is defined as \(a_n = (-1)^n(a_{n - 1} + a_{n - 2})\), where \(a_1 = 1\) and \(a_2 = 1\). What is the value of the sum \(a_{35} + a_{38}\) ?
(A) -1
(B) 0
(C) 1
(D) 1597
(E) 2584
(A) -1
(B) 0
(C) 1
(D) 1597
(E) 2584
09 Sep 2015, 02:11
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Bunuel
Solution : \(a_{38} = (-1)^{38}(a_{38 - 1} + a_{38 - 2}) = a_{37} + a_{36} = (-1)^{37}(a_{37 - 1} + a_{37 - 2}) + a_{36} = - a_{36}- a_{35}+ a_{36} = - a_{35}\)
Therefore, \(a_{38} + a_{35} = 0\)
Option B
General Discussion
13 Sep 2015, 09:53
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Bunuel
MANHATTAN GMAT OFFICIAL SOLUTION:
The first step in solving this problem is to figure out the first several terms of S, after the first two (which equal 1):
\(a_3 = (-1)^3(a_2 + a_1) = (-1)(1 + 1) = -2\)
\(a_4 = (-1)^4(a_3 + a_2) = (1)(-2 + 1) = -1\)
\(a_5 = (-1)^5(a_4 + a_3) = (-1)(-1 + -2) = 3\)
\(a_6 = (-1)^6(a_5 + a_4) = (1)(3 + -1) = 2\)
\(a_7 = (-1)^7(a_6 + a_5) = (-1)(3 + 2) = -5\)
\(a_8 = (-1)^8(a_7 + a_8) = (1)(-5 + 2) = -3\)
Unfortunately, the terms do not seem to be repeating, but we might notice that each even term is equal to the negative of the term 3 positions prior.
This would mean that \(a_35 + a_38\) would equal zero.
Let’s test this hypothesis by working backwards from the expression. Note that each even term has a positive 1 in front as a factor, whereas each odd term has -1 as a factor.
\(a_35 + a_38\)
\(= a_35 + (a_37 + a_36)\)
\(= a_35 + ((-a_36 – a_35)+ a_36)\)
= 0
Incidentally, the name of the problem is inspired by the famous Fibonacci sequence, in which each term after the first two is the sum of the two previous terms (and terms #1 and #2 are both equal to 1): 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc.
The alternating sign in the definition of our sequence throws a monkey wrench into the works, but in many respects the sequence in this problem is Fibonacci-like: all of the terms have values equal to Fibonacci numbers or the negatives of Fibonacci numbers.
The correct answer is B.
