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In the sequence of positive numbers x1, x2, x3, ..., what is the value 21 Jun 2010, 15:58
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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) \(x_5=\frac{x_4}{x_4+1}\)

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Bunuel
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In the sequence of positive numbers x1, x2, x3, ..., what is the value 21 Jun 2010, 16:33
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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 (2):

    \(x_5=\frac{x_4}{2}\).

From (1):

    \(x_5=\frac{x_4}{2}\).

Thus:

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

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

    \(x_4=1\).

Since from (1) \(x_i=\frac{x_{(i-1)}}{2}\), then:

    \(x_4=\frac{x_3}{2}=\frac{x_2}{4}=\frac{x_1}{8}\).

So:

    \(x_4=\frac{x_1}{8}\);

    \(1=\frac{x_1}{8}\);

    \(x_1=8\).


Sufficient.

Answer: C.
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In the sequence of positive numbers x1, x2, x3, ..., what is the value 20 Jan 2014, 10:39
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12. Sequences



For other subjects:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread

Hope it helps.
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In the sequence of positive numbers x1, x2, x3, ..., what is the value 22 Jun 2010, 11:45
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why is it so that we are doing the (1/2) 3 times?
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In the sequence of positive numbers x1, x2, x3, ..., what is the value 22 Jun 2010, 12:34
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ruchichitral
why is it so that we are doing the (1/2) 3 times?
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\(x_i=\frac{x_{(i-1)}}{2}\), 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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In the sequence of positive numbers x1, x2, x3, ..., what is the value 24 Aug 2010, 06:39
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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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In the sequence of positive numbers x1, x2, x3, ..., what is the value 24 Aug 2010, 06:45
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jainsaurabh
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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Stem says: "In the sequence of positive numbers ..."
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In the sequence of positive numbers x1, x2, x3, ..., what is the value 24 Aug 2010, 06:49
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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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In the sequence of positive numbers x1, x2, x3, ..., what is the value 24 Oct 2013, 07:01
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abhisheksharma85
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..
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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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In the sequence of positive numbers x1, x2, x3, ..., what is the value 26 Oct 2013, 14:30
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Bunuel
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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last part is not clear. (1/2)^3 how did u get it?
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In the sequence of positive numbers x1, x2, x3, ..., what is the value 27 Oct 2013, 07:00
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Raihanuddin
Bunuel
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?
Show more

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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In the sequence of positive numbers x1, x2, x3, ..., what is the value 06 Jul 2014, 09:51
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Bunuel
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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Bunuel: to get x4 = 1 do we cross multiply? Can you show the steps to attain this value ? :)
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In the sequence of positive numbers x1, x2, x3, ..., what is the value 06 Jul 2014, 12:10
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sagnik242
Bunuel
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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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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In the sequence of positive numbers x1, x2, x3, ..., what is the value 11 Oct 2018, 22:10
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Bunuel
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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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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In the sequence of positive numbers x1, x2, x3, ..., what is the value 11 Oct 2018, 23:21
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Bunuel
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 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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Yes, all terms in a sequence can be the same. For example, {1, 1, 1, 1, ...}.
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In the sequence of positive numbers x1, x2, x3, ..., what is the value 13 Apr 2021, 10:40
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Bunuel
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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How do we know its a GP question?
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In the sequence of positive numbers x1, x2, x3, ..., what is the value 13 Apr 2021, 10:44
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ManyataM
Bunuel
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.

How do we know its a GP question?
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\(x_i=\frac{x_{(i-1)}}{2}\) means that each term is 1/2 of the previous term so we have geometric progression with common ratio of 1/2.
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In the sequence of positive numbers x1, x2, x3, ..., what is the value 26 Aug 2025, 09:52
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you are cancelling out two terms but what if X4 =0 than there will be two solutions so E option will be the answer
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In the sequence of positive numbers x1, x2, x3, ..., what is the value 26 Aug 2025, 10:09
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iamdarshitgupta
you are cancelling out two terms but what if X4 =0 than there will be two solutions so E option will be the answer
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You should read the question and the discussion more carefully:

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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In the sequence of positive numbers x1, x2, x3, ..., what is the value 01 Mar 2026, 19:19
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testprep2010
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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C




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