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

Question

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} 2022-09-07_16-53-13.png

Bunuel · Accepted Answer

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.

Bunuel · Answer

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Hope it helps.

ruchichitral · Answer

why is it so that we are doing the (1/2) 3 times?

Bunuel · Answer

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 .

jainsaurabh · Answer

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.

Bunuel · Answer

Stem says: "In the sequence of  positive numbers  ..."

jainsaurabh · Answer

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.

Chiranjeevee · Answer

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

we can only know

X 2  = X 1  / 2 or
X 3  = X 2  / 2 = X 1  / 4 or
X 4  = X 1  / 8 or
X 5  = X 1  / 16 etc....
insufficient

from 2, X 5  = X 4  / X 4 +1
cannot find X 1  , insufficient

combine 1+2, X 5  = X 1  / 16
and X 4  = X 1  / 8

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

can find X 1  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

Raihanuddin · Answer

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

Bunuel · Answer

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.

sagnik242 · Answer

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

Bunuel · Answer

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.

sandman13 · Answer

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?

Bunuel · Answer

Yes, all terms in a sequence can be the same. For example, {1, 1, 1, 1, ...}.

ManyataM · Answer

How do we know its a GP question?

Bunuel · Answer

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.

dotsoftme · Answer

you are cancelling out two terms but what if X4 =0 than there will be two solutions so E option will be the answer

Bunuel · Answer

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}

totaltestprepNick · Answer

C https://www.youtube.com/watch?v=Hinqg32wQR4 Nick Slavkovich, GMAT/GRE tutor with 20+ years of experience tutoring@totaltestprep.net

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

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

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