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va95
Joined: 24 Dec 2016
Last visit: 19 Jun 2025
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Location: Armenia
Concentration: Statistics
Schools: Duke Fuqua HBS '26 Wharton '26
GMAT 1: 720 Q49 V40
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Updated on: 23 Jun 2017, 09:45
Originally posted by va95 on 23 Jun 2017, 08:59.
Last edited by mikemcgarry on 23 Jun 2017, 09:45, edited 2 times in total.
Last edited by mikemcgarry on 23 Jun 2017, 09:45, edited 2 times in total.
Edited the question.
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
37% (02:33) correct
63%
(02:28)
wrong
based on 132
sessions
History
Date
Time
Result
Not Attempted Yet
In the sequence \(a_1 = 2017\) and \(a_n=\frac{a_{n-1} - 1}{a_{n-1}}\), what is the value of \(a_{2017}\)?
(A) \(-2017\)
(B) \(\frac{-1}{2017}\)
(C) \(\frac{2016}{2017}\)
(D) 1
(E) 2017
(A) \(-2017\)
(B) \(\frac{-1}{2017}\)
(C) \(\frac{2016}{2017}\)
(D) 1
(E) 2017
23 Jun 2017, 09:59
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Vardan95
Dear Vardan95,
I'm happy to respond.
This is a diabolically hard question, extremely clever! I don't know if this would appear on the GMAT--it really would be at about the outer level of what the GMAT would ask. Here's my solution.
\(a_1 = 2017\)
\(a_2 = \dfrac{a_1 - 1}{a_1} = \dfrac{2016}{2017}\)
\(a_3= \dfrac{a_2 - 1}{a_2} = \dfrac{-1/2017}{2016/2017} = \dfrac{-1}{2016}\)
\(a_4 = \dfrac{a_3 - 1}{a_3} = \dfrac{-2017/2016}{-1/2016} = 2017\)
BINGO! That's the key to the problem! The sequence is cyclical, repeating with a cycle of three. For every n that is a multiple of 3, \(a_n = \dfrac{-1}{2016}\) and the following term will be \(a_{n+1} = 2017\).
Well, 2016 is a multiple of three, because 2 + 0 + 1 + 6 = 9, a multiple of three. Thus we know
\(a_{2016} = \dfrac{-1}{2016}\)
\(a_{2017} = 2017\)
Answer = (E)
A fantastic problem! Does this make sense?
Mike
General Discussion
va95
Joined: 24 Dec 2016
Last visit: 19 Jun 2025
Posts: 60
Own Kudos:
Given Kudos: 153
Location: Armenia
Concentration: Statistics
Schools: Duke Fuqua HBS '26 Wharton '26
GMAT 1: 720 Q49 V40
GMAT 2: 770 Q50 V47
GPA: 3.4
WE:Consulting (Consulting)
Products:
23 Jun 2017, 10:15
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Dear Mike,
Great explanation, this question is from 12 grade school Olimpiad. It was in the hardest section.
Posted from my mobile device
Posted from my mobile device
Great explanation, this question is from 12 grade school Olimpiad. It was in the hardest section.
Posted from my mobile device
Posted from my mobile device
