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

Question

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 .

D3N0 · Answer

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

Ans: D

Solution: put the value and you will get S4= +11 , -9
But as we know that root of -ve is not possible so S4= +11
Which gives us S3= √11 + 1

kunal555 · Answer

S_n=(S_{n-1}-1)^2 
 S_5=(S_{5-1}-1)^2 
 100=(S_4-1)^2 
 S_4 = 11

S_4=(S_{3-1}-1)^2 
 11=(S_3-1)^2 
 \sqrt{11} = S_3 - 1 
 S_3 = \sqrt{11}+1

Answer:- D

Bunuel · Answer

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

2) add 1

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

The correct answer is D.

RR88 · Answer

S5 = (S4 - 1)^2 = ((S3-1)^2 - 1)^2 =100

S3 = \sqrt{11} + 1

bumpbot · Answer

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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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