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# The sequence S_n is such that, for every n where n > 1, S_n=(S_{n-1}-1

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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:

$$11 = (S_3-1)^2$$

$$\sqrt{11} = S_3 – 1$$

$$S_3 = \sqrt{11} + 1$$

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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
The text passage contains a mathematical question and provides an answer to it. The question is about finding the value of \( S_3 \) in a sequence given the value of \( S_5 \). The answer is provided in the text as \( \sqrt{11} + 1 \).
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
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# The sequence S_n is such that, for every n where n > 1, S_n=(S_{n-1}-1

SORT BY:
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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
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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
user
The text passage contains a mathematical question and provides an answer to it. The question is about finding the value of \( S_3 \) in a sequence given the value of \( S_5 \). The answer is provided in the text as \( \sqrt{11} + 1 \).

TYPE: SINGLE_QUESTION_WITH_ANSWER