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

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

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06 Jul 2014, 23:57
ravih wrote:
Bunnuel,

Are AP and GP formulas and concepts required for GMAT? Its obvious that knowing them can be helpful like it helped here, but does GMAT need you to know these thoroughly?

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

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03 Aug 2015, 05:31
We can't assume x5=$$\frac{x4}{(x4+1)}$$ means, x4=$$\frac{x3}{(x3+1)}$$ and x3=$$\frac{x2}{(x2+1)}$$ and so on?
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In the sequence of positive numbers x1, x2, x3, ..., what  [#permalink]

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03 Aug 2015, 05:34
visram04 wrote:
We can't assume x5=$$\frac{x4}{(x4+1)}$$ means, x4=$$\frac{x3}{(x3+1)}$$ and x3=$$\frac{x2}{(x2+1)}$$ and so on?

No, definitely not, unless

1. It is mentioned in the stem
2. You can calculate that particular relation.

The equation you have mentioned between $$x_4$$ and $$x_3$$ can not be proved for both the points mentioned above.

Case in point, once you calculate that $$x_1 = 8$$

$$x_2 = 4$$
$$x_3 = 2$$
$$x_4 = 1$$
$$x_5 = 0.5$$

So, $$x_4 \neq \frac{x_3}{x_3+1}$$

Furthermore, for your statement to hold true, the question should have mentioned

$$x_i \neq \frac{x_i}{x_i+1}$$

instead of a particular value relation (example, $$x_4$$ and $$x_5$$)

Also, if ever you find that the solutions you get after solving 2 statements separately are contradictory, then you have made some mistake as per GMAT's guidelines, the 2 statements provided can not contradict each other.

Hope this helps.
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In the sequence of positive numbers x1, x2, x3, ..., what  [#permalink]

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17 Jan 2016, 05:27
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.
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Re: In the sequence of positive numbers x1, x2, x3, ..., what  [#permalink]

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16 Jul 2016, 00:44
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

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

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19 May 2018, 20:40
Is my eye sight weak? Or is the image too bad? I couldn't recognise 5 on x5. It looked like xs.
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Re: In the sequence of positive numbers x1, x2, x3, ..., what  [#permalink]

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11 Oct 2018, 21:10
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?
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Re: In the sequence of positive numbers x1, x2, x3, ..., what  [#permalink]

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11 Oct 2018, 22:21
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, ...}.
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Re: In the sequence of positive numbers x1, x2, x3, ..., what  [#permalink]

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18 Nov 2018, 14:12
Why are we allowed to apply the information found in 1) to the question stem?

The sequence in part 1) is defined for all integers i>1, yet in this case the question stem wants us to find the value of 1?

For this reason i thought the answer should be E).

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

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24 Nov 2018, 16:57
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.

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In the sequence of positive numbers x1, x2, x3, ..., what &nbs [#permalink] 24 Nov 2018, 16:57

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