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In the sequence of positive numbers x1, x2, x3, ..., what

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In the sequence of positive numbers x1, x2, x3, ..., what [#permalink] New post 21 Jun 2010, 14:58
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In the sequence of positive numbers x_1, x_2, x_3, ..., what is the value of x_1?

(1) x_i=\frac{x_{(i-1)}}{2} for all integers i>1.

(2) x_5=\frac{x_4}{x_4+1}
[Reveal] Spoiler: OA
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Re: GMAT Prep Question 2 [#permalink] New post 21 Jun 2010, 15:33
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In the sequence of positive numbers x_1, x_2, x_3, ..., what is the value of x_1?

(1) x_i=\frac{x_{(i-1)}}{2} for all integers i>1 --> we have the general formula connecting two consecutive terms (basically we have geometric progression with common ratio 1/2), but without the value of any term this info is insufficient to find x_1.

(2) x_5=\frac{x_4}{x_4+1} --> we have the relationship between x_5 and x_4, also insufficient to find x_1 (we cannot extrapolate the relationship between x_5 and x_4 to all consecutive terms in the sequence).

(1)+(2) From (1) x_5=\frac{x_4}{2} --> \frac{x_4}{2}=\frac{x_4}{x_4+1} --> x_4=1 --> x_4=1=x_1*(\frac{1}{2})^3 --> x_1=8. Sufficient.

Answer: C.
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Re: GMAT Prep Question 2 [#permalink] New post 22 Jun 2010, 10:45
why is it so that we are doing the (1/2) 3 times?
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Re: GMAT Prep Question 2 [#permalink] New post 22 Jun 2010, 11:34
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ruchichitral wrote:
why is it so that we are doing the (1/2) 3 times?


x_i=\frac{x_{(i-1)}}{2}, so every next term is preivious term times \frac{1}{2} --> x_4=x_3*\frac{1}{2}=x_2*\frac{1}{2}*\frac{1}{2}=x_1*\frac{1}{2}*\frac{1}{2}*\frac{1}{2}=x_1*(\frac{1}{2})^3.
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Re: GMAT Prep Question 2 [#permalink] New post 24 Aug 2010, 05:39
Thanks Bunuel,

But how do you account for the fact that x4 could be equal to zero.

By taking both the statements together, one of the solutions is also x4 = 0. It nowhere mentions in the question that the sequence has all distinct numbers. Or may be I am unaware that sequence is meant to consist of distinct numbers only.
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Re: GMAT Prep Question 2 [#permalink] New post 24 Aug 2010, 05:45
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jainsaurabh wrote:
Thanks Bunuel,

But how do you account for the fact that x4 could be equal to zero.

By taking both the statements together, one of the solutions is also x4 = 0. It nowhere mentions in the question that the sequence has all distinct numbers. Or may be I am unaware that sequence is meant to consist of distinct numbers only.


Stem says: "In the sequence of positive numbers ..."
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Re: GMAT Prep Question 2 [#permalink] New post 24 Aug 2010, 05:49
Oooops !!! missed that one.

I must say Bunuel you are a champion! I follow your posts and sometimes I am just blown away by your answers.
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Re: In the sequence of positive numbers x1, x2, x3, ..., what [#permalink] New post 04 Jun 2013, 04:06
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Re: In the sequence of positive numbers x1, x2, x3, ..., what [#permalink] New post 05 Jun 2013, 03:38
1 st.) Does not tell us anything. It only tells us the sequence formula and how each term in the sequence are related, but no numbers.

2 st.) Again this statement tells us the relation of X5 and X4, no real numbers. Not sufficient.

Combining two statements we see that according to the first formula X4+1=2 which means that X4=1. Solving it for X3=X2/2, X4=X2/4--->1=X2/4---> X2=4. X2=X1/2--->X1=8

The answer is C.
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Re: In the sequence of positive numbers [#permalink] New post 24 Oct 2013, 06:01
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In the sequence of positive numbers X1, X2, X3, ..... What is the value of X1

(1) Xi = Xi-1 / 2 for all integers i > 1

(2) X5 = X4 / X4+1



Guys, I need to know how to solve this question.. Thanks..



from 1, simply put the values of i = 2,3 or 4 but we can not find the value of X1

we can only know

X2 = X1 / 2 or
X3 = X2 / 2 = X1 / 4 or
X4 = X1 / 8 or
X5 = X1 / 16 etc....
insufficient

from 2, X5 = X4 / X4+1
cannot find X1 , insufficient

combine 1+2, X5 = X1 / 16
and X4 = X1 / 8

so, X1 / 16 = (X1 / 8)/((X1 / 8)+1)

can find X1 hence sufficient

Note : you will find two values of X1 from above quadratic equation i.e X1 = 0 and X1 = 8,
since it is given that X1 is positive so we cant take X1=0 hence sufficient
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Re: GMAT Prep Question 2 [#permalink] New post 26 Oct 2013, 13:30
Bunuel wrote:
In the sequence of positive numbers x_1, x_2, x_3, ..., what is the value of x_1?

(1) x_i=\frac{x_{(i-1)}}{2} for all integers i>1 --> we have the general formula connecting two consecutive terms (basically we have geometric progression with common ratio 1/2), but without the value of any term this info is insufficient to find x_1.

(2) x_5=\frac{x_4}{x_4+1} --> we have the relationship between x_5 and x_4, also insufficient to find x_1 (we cannot extrapolate the relationship between x_5 and x_4 to all consecutive terms in the sequence).

(1)+(2) From (1) x_5=\frac{x_4}{2} --> \frac{x_4}{2}=\frac{x_4}{x_4+1} --> x_4=1 --> x_4=1=x_1*(\frac{1}{2})^3 --> x_1=8. Sufficient.

Answer: C.

last part is not clear. (1/2)^3 how did u get it?
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Re: GMAT Prep Question 2 [#permalink] New post 27 Oct 2013, 06:00
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Bunuel wrote:
In the sequence of positive numbers x_1, x_2, x_3, ..., what is the value of x_1?

(1) x_i=\frac{x_{(i-1)}}{2} for all integers i>1 --> we have the general formula connecting two consecutive terms (basically we have geometric progression with common ratio 1/2), but without the value of any term this info is insufficient to find x_1.

(2) x_5=\frac{x_4}{x_4+1} --> we have the relationship between x_5 and x_4, also insufficient to find x_1 (we cannot extrapolate the relationship between x_5 and x_4 to all consecutive terms in the sequence).

(1)+(2) From (1) x_5=\frac{x_4}{2} --> \frac{x_4}{2}=\frac{x_4}{x_4+1} --> x_4=1 --> x_4=1=x_1*(\frac{1}{2})^3 --> x_1=8. Sufficient.

Answer: C.

last part is not clear. (1/2)^3 how did u get it?


Please read the red part carefully.

Or from x_i=\frac{x_{(i-1)}}{2}:

x_2=\frac{x_1}{2};

x_3=\frac{x_2}{2}=\frac{x_1}{4};

x_4=\frac{x_3}{2}=\frac{x_1}{8}.

Hope it's clear.
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Re: In the sequence of positive numbers x1, x2, x3, ..., what [#permalink] New post 29 Oct 2013, 03:15
What would the difficulty level of this question be? Is it above 700 level?
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Re: In the sequence of positive numbers x1, x2, x3, ..., what [#permalink] New post 29 Oct 2013, 05:48
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Re: In the sequence of positive numbers x1, x2, x3, ..., what [#permalink] New post 21 Jan 2014, 12:21
This is what I understand thus far

testprep2010 wrote:
In the sequence of positive numbers x_1, x_2, x_3, ..., what is the value of x_1?

(1) x_i=\frac{x_{(i-1)}}{2} for all integers i>1.

(2) x_5=\frac{x_4}{x_4+1}


when combined we can infer that x_4+1 =2 thus x_4=1

and so if you can get a number for one answer, you can get a number for any answer and thus the answer is C?
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Re: In the sequence of positive numbers x1, x2, x3, ..., what [#permalink] New post 17 Jun 2014, 16:18
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I tried to do this by plugging in numbers and after 3+ minutes, I chose "Neither Suff" b/c I wasn't seeing any trends.

A and B are both insufficient because we don't have a starting point. When I combined, I still didn't see a starting point and went through the calculations only to realize that I STILL didn't have a starting point.

In retrospect, if I was to plug in numbers, what would be the better approach? Should I pick a number for say X5 and work backwards on both A and B or does picking numbers in this problem fail? It seemed to me that I could get MULTIPLE values since I could assign ANY value to X5 etc.?
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Re: In the sequence of positive numbers x1, x2, x3, ..., what [#permalink] New post 18 Jun 2014, 04:29
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Bunuel wrote:
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I tried to do this by plugging in numbers and after 3+ minutes, I chose "Neither Suff" b/c I wasn't seeing any trends.

A and B are both insufficient because we don't have a starting point. When I combined, I still didn't see a starting point and went through the calculations only to realize that I STILL didn't have a starting point.

In retrospect, if I was to plug in numbers, what would be the better approach? Should I pick a number for say X5 and work backwards on both A and B or does picking numbers in this problem fail? It seemed to me that I could get MULTIPLE values since I could assign ANY value to X5 etc.?


On DS questions when plugging numbers, goal is to prove that the statement is not sufficient. So we should try to get an YES answer with one chosen number(s) and a NO with another.

You can easily see that (1) and (2) are not sufficient alone: different numbers plugged there will lead to different values of x1. When you take the statements together you are able to find the value of x4, and then the value of x1, so no need to plug-in there.
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Re: In the sequence of positive numbers x1, x2, x3, ..., what [#permalink] New post 06 Jul 2014, 08:51
Bunuel wrote:
In the sequence of positive numbers x_1, x_2, x_3, ..., what is the value of x_1?

(1) x_i=\frac{x_{(i-1)}}{2} for all integers i>1 --> we have the general formula connecting two consecutive terms (basically we have geometric progression with common ratio 1/2), but without the value of any term this info is insufficient to find x_1.

(2) x_5=\frac{x_4}{x_4+1} --> we have the relationship between x_5 and x_4, also insufficient to find x_1 (we cannot extrapolate the relationship between x_5 and x_4 to all consecutive terms in the sequence).

(1)+(2) From (1) x_5=\frac{x_4}{2} --> \frac{x_4}{2}=\frac{x_4}{x_4+1} --> x_4=1 --> x_4=1=x_1*(\frac{1}{2})^3 --> x_1=8. Sufficient.

Answer: C.


Bunuel: to get x4 = 1 do we cross multiply? Can you show the steps to attain this value ? :)
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Re: In the sequence of positive numbers x1, x2, x3, ..., what [#permalink] New post 06 Jul 2014, 11:10
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Bunuel wrote:
In the sequence of positive numbers x_1, x_2, x_3, ..., what is the value of x_1?

(1) x_i=\frac{x_{(i-1)}}{2} for all integers i>1 --> we have the general formula connecting two consecutive terms (basically we have geometric progression with common ratio 1/2), but without the value of any term this info is insufficient to find x_1.

(2) x_5=\frac{x_4}{x_4+1} --> we have the relationship between x_5 and x_4, also insufficient to find x_1 (we cannot extrapolate the relationship between x_5 and x_4 to all consecutive terms in the sequence).

(1)+(2) From (1) x_5=\frac{x_4}{2} --> \frac{x_4}{2}=\frac{x_4}{x_4+1} --> x_4=1 --> x_4=1=x_1*(\frac{1}{2})^3 --> x_1=8. Sufficient.

Answer: C.


Bunuel: to get x4 = 1 do we cross multiply? Can you show the steps to attain this value ? :)


Sure.

\frac{x_4}{2}=\frac{x_4}{x_4+1};

Reduce by x4: \frac{1}{2}=\frac{1}{x_4+1};

Cross-multiply: x_4+1=2 --> x_4=1.

Hope it's clear.
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Re: In the sequence of positive numbers x1, x2, x3, ..., what [#permalink] New post 06 Jul 2014, 19:39
Bunnuel,

Are AP and GP formulas and concepts required for GMAT? Its obvious that knowing them can be helpful like it helped here, but does GMAT need you to know these thoroughly?
Re: In the sequence of positive numbers x1, x2, x3, ..., what   [#permalink] 06 Jul 2014, 19:39
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