Data Sufficiency

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)

Qs is from GMAT Prep. Please help in answering

DevB 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)

Statement 1: x(i) = [x(i -1)]/2.
In other words, each term is 1/2 the value of the preceding term.
Since x� can be equal to any positive value, INSUFFICIENT.

Statement 2: xâ‚… = xâ‚„/(xâ‚„ + 1)
Here, only the relationship between xâ‚… and xâ‚„ is known.
Thus, x� can be equal to any positive value.
INSUFFICIENT.

Statements combined:
Statement 1 indicates that each term is 1/2 the preceding term, implying that xâ‚… = xâ‚„/2.
Statement 2 indicates that xâ‚… = xâ‚„/(xâ‚„ + 1).

Since the two expressions in red are equal to the same value, we get:
xâ‚„/2 = xâ‚„/(xâ‚„ + 1).

Cross-multiplying, we get:
(xâ‚„)(xâ‚„ + 1) = 2xâ‚„

Since all of the values in the sequence are positive, we can safely divide each side by xâ‚„, yielding the following:
xâ‚„ + 1 = 2
xâ‚„ = 1.

Since each term is 1/2 the preceding term, we get:
x₄ = 1, x₃ = 2, x₂ = 4, x� = 8.
SUFFICIENT.

The correct answer is C.

DevB 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)

Qs is from GMAT Prep. Please help in answering

Question : Value of x4 = ?

Statement 1) xi = x(i -1)/2.
None of the terms of the Series is known to relate with the given function, therefore
INSUFFICIENT

Statement 2) xâ‚… = xâ‚„/(xâ‚„ + 1)
Relationship between two consecutive terms is given but NONE OF THE TERMS is given therefore
INSUFFICIENT.

Combining the two statements
xâ‚… = xâ‚„/(xâ‚„ + 1) AND xi = x(i -1)/2

i.e. xâ‚… = xâ‚„/(xâ‚„ + 1) AND x5 = x(4)/2
i.e. (xâ‚„ + 1)=2
i.e. xâ‚„ = 1
Now we have the relationship between two consecutive terms and using that relationship every terms of the Series can be Calculated, therefore,
SUFFICIENT

Answer: Option C

Statement 1: x(i) = [x(i -1)]/2.
In other words, each term is 1/2 the value of the preceding term.
Since x� can be equal to any positive value, INSUFFICIENT.

Hi GMATGuruNY ,

Can you please explain more about statement 1 , because I was putting the value of i, which is greater than 1 and every time i got -1.

Please explain sir.

Many thanks in advance.

SJ

jain2016 wrote:

Statement 1: x(i) = [x(i -1)]/2.
In other words, each term is 1/2 the value of the preceding term.
Since x� can be equal to any positive value, INSUFFICIENT.

Hi GMATGuruNY ,

Can you please explain more about statement 1 , because I was putting the value of i, which is greater than 1 and every time i got -1.

Please explain sir.

Many thanks in advance.

SJ

SJ

I am also confused with thee question you have posted. What is the source of this question? I am asking because if the term in statement 1 is JUST : Xi=(Xi-1)/2 where i is the subscript and -1 is actual INTEGER and not the part of subscript, then you will always get -1.

BUT if the term is such that the i and (i-1) are subscripts then above explained answers are correct.

Please check what is written in the actual problem.

Regards
Nits

jain2016 wrote:

Statement 1: x(i) = [x(i -1)]/2.
In other words, each term is 1/2 the value of the preceding term.
Since x� can be equal to any positive value, INSUFFICIENT.

Hi GMATGuruNY ,

Can you please explain more about statement 1 , because I was putting the value of i, which is greater than 1 and every time i got -1.

Please explain sir.

Many thanks in advance.

SJ

Hi Experts ,

Please explain,, where am I missing.

Many thanks in advance

SJ

jain2016 wrote:

Statement 1: x(i) = [x(i -1)]/2.
In other words, each term is 1/2 the value of the preceding term.
Since x� can be equal to any positive value, INSUFFICIENT.

Hi GMATGuruNY ,

Can you please explain more about statement 1 , because I was putting the value of i, which is greater than 1 and every time i got -1.

Please explain sir.

Many thanks in advance.

SJ

Statement 1: x(i) = [x(i -1)]/2
Thus:
x₂ = x�/2.
Rephrased:
x� = 2(x₂).

If x₂ = 1, then x� = 2*1 = 2.
If x₂ = 2, then x� = 2*2 = 4.
Since x� can be different values, INSUFFICIENT.

GMATGuruNY wrote:

jain2016 wrote:

Statement 1: x(i) = [x(i -1)]/2.
In other words, each term is 1/2 the value of the preceding term.
Since x� can be equal to any positive value, INSUFFICIENT.

Hi GMATGuruNY ,

Can you please explain more about statement 1 , because I was putting the value of i, which is greater than 1 and every time i got -1.

Please explain sir.

Many thanks in advance.

SJ

Statement 1: x(i) = [x(i -1)]/2
Thus:
x₂ = x�/2.
Rephrased:
x� = 2(x₂).

If x₂ = 1, then x� = 2*1 = 2.
If x₂ = 2, then x� = 2*2 = 4.
Since x� can be different values, INSUFFICIENT.

Hi GMATGuruNY ,

Thank you sir. I got it.

Thanks,

SJ

DevB 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)

Qs is from GMAT Prep. Please help in answering

Old post, but here's how I would approach it.

(1) is clearly insufficient on its own. We now have a formula to calculate xi, but only when we know the preceding number in the sequence. Right now, we don't know any values in the sequence.

(2) is also insufficient on its own. It gives us an alternate way to calculate x5. However, it still requires us to know the value of x4. Since we do not know the value of x4 and cannot calculate it from this equation alone, this is insufficient to answer the target question.

Now combining these, we have two separate ways to calculate x5.

Using (1), x5 = x4 / 2.

Using (2), x5 = x4 / (x4 + 1).

We can set these two equations equal to one another and solve for x4.

x4 / 2 = x4 / (x4 + 1). A quick shortcut would be to notice that the numerators are the same, so the denominators must be equal. Thus, x4 + 1 must equal 2 and so x4 must equal 1.

We can now use the value of x4 to work back to calculate x1 if we wanted to. However, this would be a waste of time since we are only concerned with whether we could evaluate x1. Since we can definitely find a value for x1, together the two statements are sufficient. Therefore, the answer is C.