text: Last visit was: 14 Jul 2024, 10:32 It is currently 14 Jul 2024, 10:32
  Toolkit
  GMAT Club Daily Prep
  Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
  
  Customized
  for You
  
  we will pick new questions that match your level based on your Timer History
  
  Track
  
  every week, we’ll send you an estimated GMAT score based on your performance
  
  Practice
  Pays
  
  we will pick new questions that match your level based on your Timer History
  Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
  
  # The sequence S_n is such that, for every n where n > 1, S_n=(S_{n-1}-1
  
  SORT BY:
  Tags:
  Show Tags
  Hide Tags
  Math Expert
  Joined: 02 Sep 2009
  Posts: 94342
  Own Kudos [?]: 640753 [2]
  Given Kudos: 85011
  Senior Manager
  Joined: 21 Jan 2015
  Posts: 422
  Own Kudos [?]: 359 [1]
  Given Kudos: 82
  Location: India
  Concentration: Strategy, Marketing
  GMAT 1: 620 Q48 V28
  GMAT 2: 690 Q49 V35
  WE:Sales (Consumer Products)
  Manager
  Joined: 29 Jul 2015
  Posts: 144
  Own Kudos [?]: 653 [1]
  Given Kudos: 59
  Math Expert
  Joined: 02 Sep 2009
  Posts: 94342
  Own Kudos [?]: 640753 [0]
  Given Kudos: 85011
  The sequence S_n is such that, for every n where n > 1, S_n=(S_{n-1}-1 [#permalink]
  Bunuel wrote:
  The sequence $$S_n$$ is such that, for every n where n > 1, $$S_n=(S_{n-1}-1)^2$$. If $$S_5 = 100$$, what is $$S_3$$?
  
  (A) 10
  (B) 11
  (C) √11
  (D) √11 + 1
  (E) √101
  
  Kudos for a correct solution.
  
  MANHATTAN GMAT OFFICIAL SOLUTION:
  
  This problem is much easier if we can “decode” the sequence $$S_n=(S_{n-1}-1)^2$$
  
  In other words, $$S_n$$ is just any term in the sequence (for instance, $$S_3$$ would be the 3rd term, $$S_10$$ would be the 10th term, etc.)
  
  $$S_{n-1}$$ just means the term that comes right before $$S_n$$.
  
  Therefore, we can rephrase $$S_n=(S_{n-1}-1)^2$$ as “To get any term, take the term before it, subtract one, then square.”
  
  In this problem, however, we’ve been given $$S_5$$ and we want $$S_3$$. That is, we must go backwards.
  
  To “go backwards” in a sequence, do the opposite of each step, in the opposite order. That is, if, to go from $$S_1$$ to $$S_2$$ you would:
  
  1) subtract 1
  
  2) square
  
  …then to go backwards, you would:
  
  1) square root
  
  So, if $$S_5 = 100$$, square root to get 10 and add 1 to get 11. Notice that we do not have to worry about the possibility of a negative square root, because every term is the square of some number, so no term can be negative.
  
  If $$S_4 = 11$$, square root to get √11 and add 1 to get √11 + 1. The answer is D.
  
  The problem could also be solved a bit more algebraically as follows:
  
  $$S_n=(S_{n-1}-1)^2$$
  
  $$S_5 = 100$$
  
  Therefore:
  
  $$100 = (S_4-1)^2$$
  
  $$10 = S_4 - 1$$ (Again, we drop the possibility of a negative root, because $$S_4$$ itself is a square.)
  
  $$S_4 = 11$$
  
  Now repeat the process:
  
  $$11 = (S_3-1)^2$$
  
  $$\sqrt{11} = S_3 – 1$$
  
  $$S_3 = \sqrt{11} + 1$$
  
  Manager
  Joined: 18 Oct 2016
  Posts: 112
  Own Kudos [?]: 142 [0]
  Given Kudos: 91
  Location: India
  WE:Engineering (Energy and Utilities)
  Re: The sequence S_n is such that, for every n where n > 1, S_n=(S_{n-1}-1 [#permalink]
  Bunuel wrote:
  The sequence $$S_n$$ is such that, for every n where n > 1, $$S_n=(S_{n-1}-1)^2$$. If $$S_5 = 100$$, what is $$S_3$$?
  
  (A) 10
  (B) 11
  (C) √11
  (D) √11 + 1
  (E) √101
  
  Kudos for a correct solution.
  
  $$S5 = (S4 - 1)^2 = ((S3-1)^2 - 1)^2 =100$$
  
  $$S3 = \sqrt{11} + 1$$
  Re: The sequence S_n is such that, for every n where n > 1, S_n=(S_{n-1}-1 [#permalink]
  Moderator:
  Math Expert
  94342 posts