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Manager  Joined: 03 Feb 2010
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In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X  [#permalink]

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Difficulty:   75% (hard)

Question Stats: 62% (02:43) correct 38% (02:43) wrong based on 1315 sessions

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In the infinite sequence A, $$A_n = x^{(n-1)} + x^n + x^{(n+1)} + x^{(n+2)} + x^{(n+3)}$$ where x is a positive integer constant. For what value of n is the ratio of $$A_n$$ to $$x(1+x(1+x(1+x(1+x))))$$ equal to x^5?

A. 8
B. 7
C. 6
D. 5
E. 4
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Re: In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X  [#permalink]

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ksharma12 wrote:
18. In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X^(n+3)
where x is a positive integer constant. For what value of n is the ratio of An to
x(1+x(1+x(1+x(1+x)))) equal to X^5?

(A) 8

(B) 7

(C) 6

(D) 5

(E) 4

note: An= A sub n

Can you explain this in detail? I tried expanding out the bottom equation and solving for X to equal x^5. Didnt really work out...

$$x^5=\frac{x^{(n-1)}+x^n+x^{(n+1)}+x^{(n+2)}+x^{(n+3)}}{x(1+x(1+x(1+x(1+x))))}$$

--> $$x^6(1+x(1+x(1+x(1+x))))=x^{(n-1)}+x^n+x^{(n+1)}+x^{(n+2)}+x^{(n+3)}$$

--> take $$x^{(n-1)}$$ out of the brackets

--> $$x^6(1+x(1+x(1+x(1+x))))=x^{(n-1)}(1+x+x^2+x^3+x^4)$$

--> $$x^6(1+x(1+x(1+x(1+x))))=x^{(n-1)}(1+x(1+x+x^2+x^3))$$

--> $$x^6(1+x(1+x(1+x(1+x))))=x^{(n-1)}(1+x(1+x(1+x+x^2)))$$

--> $$x^6(1+x(1+x(1+x(1+x))))=x^{(n-1)}(1+x(1+x(1+x(1+x))))$$

--> $$x^6=x^{(n-1)}$$ --> $$n-1=6$$ --> $$n=7$$

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Re: In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X  [#permalink]

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18
1
Its not so hard when you realize how can you solve it, but until that, you spent half of your life.
##### General Discussion
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Re: In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X  [#permalink]

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17
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craky wrote:
Its not so hard when you realize how can you solve it, but until that, you spent half of your life.

Oh no you don't. Work smart!

$$An = x^{n-1} + x^n + x^{n+1} + x^{n+2} + x^{n+3}$$
e.g. $$A2 = x + x^2 + x^3 + x^4 + x^5$$
Notice you can only take x common out of all these terms i.e. the smallest term $$x^{n - 1}$$

If $$\frac{An}{{x(1+x(1+x(1+x(1+x))))}} = x^5$$, it means the part: (1+x(1+x(1+x(1+x)))) will get canceled from the num and den. Ignore it.
From An, you will be able to take out $$x^6$$ common so that $$\frac{x^6}{x}$$ gives you $$x^5$$
So smallest term must be $$x^6$$ i.e. $$x^{n-1}$$. Therefore, n = 7.
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Re: In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X  [#permalink]

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subhashghosh wrote:
Hi Karishma

What is the meaning of this ?

"it means the part: (1+x(1+x(1+x(1+x)))) will get canceled from the num and den."

Regards,
Subhash

$$A2 = x + x^2 + x^3 + x^4 + x^5$$

$$\frac{An}{{x(1+x(1+x(1+x(1+x))))}} = x^5$$
Since the right side of the equation is just x^5, it means the entire expression: (1+x(1+x(1+x(1+x)))) should get canceled out which means we will get the same expression in the numerator as well. You don't need to do it. It is logical since otherwise, you will not get the reduced expression x^5. Also, you can see that you will get something like this in the numerator since the powers are increasing.

If you want to see it:
$$A2= x( 1 + x + x^2 + x^3 + x^4) = x( 1 + x(1 + x + x^2 + x^3)) = x( 1 + x(1 + x( 1 + x + x^2))) = x( 1 + x(1 + x( 1 + x( 1 + x))))$$

Similarly A7 $$= x^6( 1 + x(1 + x( 1 + x( 1 + x))))$$

So $$\frac{A7}{{x(1+x(1+x(1+x(1+x))))}} = \frac{x^6( 1 + x(1 + x( 1 + x( 1 + x))))}{x( 1 + x(1 + x( 1 + x( 1 + x))))}= x^5$$
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Re: In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X  [#permalink]

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$$A_n=x^{(n-1)}(1+x+x^2+x^3+x^4)$$
$$x(1+x(1+x(1+x(1+x))))=x(1+x+x^2+x^3+x^4)$$

$$\frac{x^{(n-1)}(1+x+x^2+x^3+x^4)}{x(1+x+x^2+x^3+x^4)}=x^5$$

$$x^{(n-1)}=x^6$$
$$n-1=6$$
$$n=7$$

Ans: "B"
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Re: In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X  [#permalink]

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rajathpanta wrote:
omg.. by the time u read and digest the question its 1 minut e

It certainly takes you a minute or even more to get through the question and digest it but after that, it takes you less than a minute to solve it. This is true for most GMAT questions. If you understand the question well, it takes you very little time to actually solve it. If you don't understand the question well, you could end up spending 20 mins on it.
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Re: In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X  [#permalink]

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great explanation. Thanks. +1 to you
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Re: In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X  [#permalink]

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mohankumarbd wrote:
Is this a typical 700 level GMAT question? or just an off topic question? experts pls advice.

Who can tell you? If you ask all those who took the test if they ever saw such a question on a real test, you might get the real picture...

IMO, the chance is slim that such a question will appear on a real test. It is too technical, too lengthy to be done with plugging in numbers...
Until now, I didn't get the feeling that GMAT wants to test just algebraic abilities. Not that this question needs some really advanced techniques, but it's above basics...
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Re: In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X  [#permalink]

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Is it a GMAT level question??????
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Re: In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X  [#permalink]

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Hi Karishma

What is the meaning of this ?

"it means the part: (1+x(1+x(1+x(1+x)))) will get canceled from the num and den."

Regards,
Subhash
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Re: In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X  [#permalink]

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I hope this is a 700+ level question...
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Re: In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X  [#permalink]

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aznboi986 wrote:
I hope this is a 700+ level question...

this is 800+ level question You definitely do not see any question like this one on real exam. GC problems usually too difficult.
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Re: In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X  [#permalink]

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i) x(1 + x(1 + x(1 + x(1 + x)))) = x + x^2 + x^3 + x^4 + x^5
ii) An = x^(n-2) (x + x^2 + x^3 + x^4 + x^5)

Divide ii by i = x^(n-2)
We want ratio = x^5. So, n-2 = 5
=> n = 7. (B)
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Re: In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X  [#permalink]

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omg.. by the time u read and digest the question its 1 minut e
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Re: In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X  [#permalink]

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Is this a typical 700 level GMAT question? or just an off topic question? experts pls advice.
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Re: In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X  [#permalink]

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mohankumarbd wrote:
Is this a typical 700 level GMAT question? or just an off topic question? experts pls advice.

It is an algebra question that looks tricky but can be easily reasoned out. It will take you some time to understand the question but once you do, you can solve it quickly - pretty much like high level GMAT questions.
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Re: In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X  [#permalink]

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I did it like it : as you can see x(1+x(1+x(1+x(1+x)))) X comes 5 times, therefore the max term will be X^5, in the question you see that you want to arrive at X^5 so it means that in the sum of X^n-1...x^n+3 the max term must be X^10 so that it can be x^5(x^5) therefore 10 = 3+n, n=7, timer indicate me 1min 53.

But definitely i had the answer, but i was unable to demonstrate it in that time, it would take more like 5 to 10 minutes.
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Re: In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X  [#permalink]

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Could someone please explain to me how that's an infinite sequence? That's what really threw me off.
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Re: In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X  [#permalink]

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mfabros wrote:
Could someone please explain to me how that's an infinite sequence? That's what really threw me off.

The information that it is an infinite sequence doesn't have much to do with the question. You are given this only to tell you that n can take any positive integer value.

An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X^(n+3) tells you that the nth term is given by plugging in the value of n in this expression. A is not a sequence of 2 or 4 terms but infinite so n can take any value. We found out that the required relation holds when n is 7. We could have just as well got n = 10298 and that would have been fine too since A has infinite terms so any value for n is alright.
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Veritas Prep GMAT Instructor Re: In the infinite sequence A, An = X^(n-1) + X^n + X^(n+1) + X^(n+2) + X   [#permalink] 19 Sep 2013, 20:48

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