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Manager  Joined: 10 Feb 2010
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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) $$x_5=\frac{x_4}{x_4+1}$$
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Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

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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.

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Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

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why is it so that we are doing the (1/2) 3 times?
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Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

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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: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

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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: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

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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: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

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Oooops !!! missed that one.

Intern  Joined: 28 Jan 2013
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Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

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abhisheksharma85 wrote:
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: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

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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.

last part is not clear. (1/2)^3 how did u get it?
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Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

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Raihanuddin wrote:
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.

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

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 is the value  [#permalink]

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Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

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Bunuel wrote:
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

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 is the value  [#permalink]

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russ9 wrote:
Bunuel wrote:
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

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 is the value  [#permalink]

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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.

Bunuel: to get x4 = 1 do we cross multiply? Can you show the steps to attain this value ? Math Expert V
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Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

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sagnik242 wrote:
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.

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 is the value  [#permalink]

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

(1) $$x_i=\frac{x_{(i-1)}}{2[/fraction (2) [m]x_5=[fraction]x_4/x_4+1}$$

This is not my strongest topic, but I just want to share my experience with this type of questions.
One can solve this type of questions if you have a general formula (sometimes given in the question stem) and concrete values for at least two other terms in the sequence.

Question: We are not given any formula here, we are just asked to find a value of X1.
(1) It's just a general formula, that gives us the relaionship of any two terms in the sequence, but we don't have any concrete values. Not sufficient
(2) Here we are given ONLY a relationship between X5 and X4, but you cannot just use this relationship for other terms, X4 and X5 could by any positive values. Not sufficient.
(1) + (2): x4=x4/x4+1 = x4/2, ok, so you can find the x4, in the same manner you can find x3 etc. so , STOP, there is no need to calculate further, one can see here, that it's possible to find x1

Would appreciate some comments from math experts. I have seen many questions of this type, and there are always about a general formula and some concrete values, which we can use to find any term in the sequence.
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GMAT 1: 750 Q49 V43 Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

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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) $$x_5=\frac{x_4}{x_4+1}$$

We need both statement
Statement 1 s giving the formula for terms greater than 1
so it cannot be used to calculate the 1st term. INSUFFICIENT

Statement 2 is giving a value for 5th term.
But it is not giving a formula to tell how that term was calculated
so cannot be used. INSUFFICIENT

Both together will yield a value for 4th term, 3term and 2nd term.
Then the 1st term can be easily calculated by observing the relationship between 3rd 4th and 5th term SUFFICIENT

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Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

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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.

Bunuel I got this wrong because I thought x1 = 0 is also a solution. But since the question says it is a sequence of positive numbers, I guess I cannot assume that.

On a slightly different note, can the values in a sequence be a constant?
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Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

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sandman13 wrote:
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.

Bunuel I got this wrong because I thought x1 = 0 is also a solution. But since the question says it is a sequence of positive numbers, I guess I cannot assume that.

On a slightly different note, can the values in a sequence be a constant?

Yes, all terms in a sequence can be the same. For example, {1, 1, 1, 1, ...}.
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Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

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A video with a more thorough explanation can be found here:

Statement (1) gives us a general equation that can be applied to any value of x. Helpful, but not sufficient.

Statement (2) gives us an equation that is specific to x5; it does not necessarily apply to every other value of x. Perhaps helpful, but not sufficient.

In combination, however, we can now write two equations with two unknowns (x5 and x4). Any time you have two unknows, you can solve for both of them if you have two equations that are different from each other (as we do here), so without doing any math we know that we can find a value for x4, and then we could go back and use the equation in statement (1) to find x1.

_________________ Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value   [#permalink] 24 Nov 2018, 17:57

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