GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 17 Nov 2019, 03:04

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

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

Author Message
TAGS:

Hide Tags

Manager
Joined: 10 Feb 2010
Posts: 130
In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

Show Tags

21 Jun 2010, 15:58
7
59
00:00

Difficulty:

35% (medium)

Question Stats:

69% (01:42) correct 31% (01:38) wrong based on 850 sessions

HideShow timer Statistics

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) $$x_5=\frac{x_4}{x_4+1}$$
Math Expert
Joined: 02 Sep 2009
Posts: 59087
Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

Show Tags

21 Jun 2010, 16:33
15
18
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.

_________________
General Discussion
Intern
Joined: 21 Jun 2010
Posts: 5
Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

Show Tags

22 Jun 2010, 11:45
1
why is it so that we are doing the (1/2) 3 times?
Math Expert
Joined: 02 Sep 2009
Posts: 59087
Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

Show Tags

22 Jun 2010, 12:34
4
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 $$\frac{1}{2}$$ --> $$x_4=x_3*\frac{1}{2}=x_2*\frac{1}{2}*\frac{1}{2}=x_1*\frac{1}{2}*\frac{1}{2}*\frac{1}{2}=x_1*(\frac{1}{2})^3$$.
_________________
Intern
Joined: 24 May 2010
Posts: 4
Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

Show Tags

24 Aug 2010, 06:39
1
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.
Math Expert
Joined: 02 Sep 2009
Posts: 59087
Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

Show Tags

24 Aug 2010, 06:45
2
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 ..."
_________________
Intern
Joined: 24 May 2010
Posts: 4
Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

Show Tags

24 Aug 2010, 06:49
2
Oooops !!! missed that one.

Intern
Joined: 28 Jan 2013
Posts: 27
Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

Show Tags

24 Oct 2013, 07:01
4
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
Manager
Joined: 11 Sep 2013
Posts: 131
Concentration: Finance, Finance
Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

Show Tags

26 Oct 2013, 14:30
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.

last part is not clear. (1/2)^3 how did u get it?
Math Expert
Joined: 02 Sep 2009
Posts: 59087
Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

Show Tags

27 Oct 2013, 07:00
1
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.
_________________
Math Expert
Joined: 02 Sep 2009
Posts: 59087
Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

Show Tags

20 Jan 2014, 10:39
7
2
Manager
Joined: 15 Aug 2013
Posts: 227
Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

Show Tags

17 Jun 2014, 17:18
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.?
Math Expert
Joined: 02 Sep 2009
Posts: 59087
Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

Show Tags

18 Jun 2014, 05:29
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.
_________________
Manager
Joined: 28 Dec 2013
Posts: 65
Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

Show Tags

06 Jul 2014, 09:51
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.

Bunuel: to get x4 = 1 do we cross multiply? Can you show the steps to attain this value ?
Math Expert
Joined: 02 Sep 2009
Posts: 59087
Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

Show Tags

06 Jul 2014, 12:10
1
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.
_________________
Senior Manager
Joined: 10 Mar 2013
Posts: 461
Location: Germany
Concentration: Finance, Entrepreneurship
Schools: WHU MBA"20 (A\$)
GMAT 1: 580 Q46 V24
GPA: 3.88
WE: Information Technology (Consulting)
Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

Show Tags

17 Jan 2016, 06:27
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.
_________________
When you’re up, your friends know who you are. When you’re down, you know who your friends are.

800Score ONLY QUANT CAT1 51, CAT2 50, CAT3 50
GMAT PREP 670
MGMAT CAT 630
KAPLAN CAT 660
Director
Joined: 04 Jun 2016
Posts: 547
GMAT 1: 750 Q49 V43
Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

Show Tags

16 Jul 2016, 01:44
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

_________________
Posting an answer without an explanation is "GOD COMPLEX". The world doesn't need any more gods. Please explain you answers properly.
FINAL GOODBYE :- 17th SEPTEMBER 2016. .. 16 March 2017 - I am back but for all purposes please consider me semi-retired.
Manager
Joined: 20 Apr 2018
Posts: 167
Concentration: Technology, Nonprofit
Schools: ISB '21 (A)
WE: Analyst (Non-Profit and Government)
Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

Show Tags

11 Oct 2018, 22:10
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?
Math Expert
Joined: 02 Sep 2009
Posts: 59087
Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

Show Tags

11 Oct 2018, 23:21
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, ...}.
_________________
Intern
Joined: 20 Aug 2018
Posts: 25
Re: In the sequence of positive numbers x1, x2, x3, ..., what is the value  [#permalink]

Show Tags

24 Nov 2018, 17:57
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

Go to page    1   2    Next  [ 23 posts ]

Display posts from previous: Sort by