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# If n is a positive integer, is the value of x - y at least

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If n is a positive integer, is the value of x - y at least  [#permalink]

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Updated on: 24 Aug 2019, 22:12
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Difficulty:

65% (hard)

Question Stats:

47% (02:14) correct 53% (02:09) wrong based on 43 sessions

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If $$n$$ is a positive integer, is the value of $$x - y$$ at least three times the value of $$5^n−3^n$$ ?

(1) $$x=5^{n-1}$$ and $$y=3^{n+1}$$

(2) $$n=2$$, $$(x+y)=32$$ and $$xy=135$$

Originally posted by chondro48 on 20 Aug 2019, 09:04.
Last edited by chondro48 on 24 Aug 2019, 22:12, edited 7 times in total.
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Re: If n is a positive integer, is the value of x - y at least  [#permalink]

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Updated on: 24 Aug 2019, 22:12
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(1) $$x-y=5^{n-1}-3^{n+1}=3*(5^n-3^n) - \frac{14}{5}*5^n$$. Since $$\frac{14}{5}*5^n > 0$$, then it must be true that $$x-y < 3*(5^n-3^n)$$.
SUFFICIENT

(2) We can calculate the value of $$(x-y)$$ from:
$$(x-y)^2= (x+y)^2-4xy$$
$$(x-y)^2= 32^2-4*135 =484$$
$$(x-y)=22$$ or $$-22$$.

Since three times the value of $$(5^2-3^2)=16$$ is always greater than both possible values of $$(x-y)$$, then it must be true that $$x-y < 3*(5^n-3^n)$$.
SUFFICIENT

+1 kudo is appreciated

Originally posted by chondro48 on 20 Aug 2019, 09:13.
Last edited by chondro48 on 24 Aug 2019, 22:12, edited 13 times in total.
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Re: If n is a positive integer, is the value of x - y at least  [#permalink]

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20 Aug 2019, 18:08
chondro48 wrote:
(1) $$x-y=5^{n-1}-3^{n+1}=3*(5^n-3^n) - \frac{14}{5}*5^n$$. Since $$\frac{14}{5}*5^n > 0$$, then it must be true that $$x-y < 3*(5^n-3^n)$$.
SUFFICIENT

(2) $$x-y=5^{n+1}-3^{n+1}=3*(5^n-3^n) + 2*5^n$$. Since $$2*5^n > 0$$, then it must be true that $$x-y > 3*(5^n-3^n)$$.
SUFFICIENT

+1 kudo is appreciated =)

Can you please explain, how you got 14/5 or *5^n in the first equation and 2*5^n in the second?
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Re: If n is a positive integer, is the value of x - y at least  [#permalink]

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Updated on: 21 Aug 2019, 11:39
1
1
arorni wrote:
chondro48 wrote:
(1) $$x-y=5^{n-1}-3^{n+1}=3*(5^n-3^n) - \frac{14}{5}*5^n$$. Since $$\frac{14}{5}*5^n > 0$$, then it must be true that $$x-y < 3*(5^n-3^n)$$.
SUFFICIENT

(2) $$x-y=5^{n+1}-3^{n+1}=3*(5^n-3^n) + 2*5^n$$. Since $$2*5^n > 0$$, then it must be true that $$x-y > 3*(5^n-3^n)$$.
SUFFICIENT

Can you please explain, how you got 14/5 or *5^n in the first equation and 2*5^n in the second?

Hi arorni,

(1)
$$x-y=5^{n-1}-3^{n+1}$$
$$=\frac{1}{5}*5^n-3*3^n$$
$$=(\frac{15}{5}*5^n-\frac{14}{5}*5^n)-3*3^n$$
$$=3*5^n-\frac{14}{5}*5^n-3*3^n$$
$$=3*5^n - 3*3^n -\frac{14}{5}*5^n$$
$$=3*(5^n-3^n) -\frac{14}{5}*5^n$$
Then, refer to my previous post for the rest.

(2)
$$x-y=5^{n+1}-3^{n+1}$$
$$=5*5^n-3*3^n$$
$$=2*5^n+3*5^n-3*3^n$$
$$=2*5^n+3*(5^n-3^n)$$
$$=3*(5^n-3^n)+2*5^n$$
Then, refer to my previous post for the rest.

Posted from my mobile device

Originally posted by chondro48 on 20 Aug 2019, 19:28.
Last edited by chondro48 on 21 Aug 2019, 11:39, edited 1 time in total.
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Re: If n is a positive integer, is the value of x - y at least  [#permalink]

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20 Aug 2019, 20:07
1
arorni wrote:
chondro48 wrote:
(1) $$x-y=5^{n-1}-3^{n+1}=3*(5^n-3^n) - \frac{14}{5}*5^n$$. Since $$\frac{14}{5}*5^n > 0$$, then it must be true that $$x-y < 3*(5^n-3^n)$$.
SUFFICIENT

(2) $$x-y=5^{n+1}-3^{n+1}=3*(5^n-3^n) + 2*5^n$$. Since $$2*5^n > 0$$, then it must be true that $$x-y > 3*(5^n-3^n)$$.
SUFFICIENT

+1 kudo is appreciated =)

Can you please explain, how you got 14/5 or *5^n in the first equation and 2*5^n in the second?

You may try number plug in method also which works smoothly if stuck in algebra.
_________________
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Re: If n is a positive integer, is the value of x - y at least  [#permalink]

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21 Aug 2019, 07:29
chondro48 wrote:
arorni wrote:
chondro48 wrote:
(1) $$x-y=5^{n-1}-3^{n+1}=3*(5^n-3^n) - \frac{14}{5}*5^n$$. Since $$\frac{14}{5}*5^n > 0$$, then it must be true that $$x-y < 3*(5^n-3^n)$$.
SUFFICIENT

(2) $$x-y=5^{n+1}-3^{n+1}=3*(5^n-3^n) + 2*5^n$$. Since $$2*5^n > 0$$, then it must be true that $$x-y > 3*(5^n-3^n)$$.
SUFFICIENT

Can you please explain, how you got 14/5 or *5^n in the first equation and 2*5^n in the second?

Hi arorni,
Please give kudos for the explanation below

(1)
$$x-y=5^{n-1}-3^{n+1}$$
$$=\frac{1}{5}*5^n-3*3^n$$
$$=(\frac{15}{5}*5^n-\frac{14}{5}*5^n)-3*3^n$$
$$=3*5^n-\frac{14}{5}*5^n-3*3^n$$
$$=3*5^n - 3*3^n -\frac{14}{5}*5^n$$
$$=3*(5^n-3^n) -\frac{14}{5}*5^n$$
Then, refer to my previous post for the rest.

(2)
$$x-y=5^{n+1}-3^{n+1}$$
$$=5*5^n-3*3^n$$
$$=2*5^n+3*5^n-3*3^n$$
$$=2*5^n+3*(5^n-3^n)$$
$$=3*(5^n-3^n)+2*5^n$$
Then, refer to my previous post for the rest.

Posted from my mobile device

Thanks a lot for the detailed explanation..
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Re: If n is a positive integer, is the value of x - y at least  [#permalink]

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21 Aug 2019, 07:33
lnm87 wrote:
arorni wrote:
chondro48 wrote:
(1) $$x-y=5^{n-1}-3^{n+1}=3*(5^n-3^n) - \frac{14}{5}*5^n$$. Since $$\frac{14}{5}*5^n > 0$$, then it must be true that $$x-y < 3*(5^n-3^n)$$.
SUFFICIENT

(2) $$x-y=5^{n+1}-3^{n+1}=3*(5^n-3^n) + 2*5^n$$. Since $$2*5^n > 0$$, then it must be true that $$x-y > 3*(5^n-3^n)$$.
SUFFICIENT

+1 kudo is appreciated =)

Can you please explain, how you got 14/5 or *5^n in the first equation and 2*5^n in the second?

You may try number plug in method also which works smoothly if stuck in algebra.

Yes..you are right...thanks for the reply, although I solved this question using numbers, I wanted to understand the equation mentioned by chondro48
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Re: If n is a positive integer, is the value of x - y at least  [#permalink]

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21 Aug 2019, 07:43
arorni wrote:

Yes..you are right...thanks for the reply, although I solved this question using numbers, I wanted to understand the equation mentioned by chondro48

Same here, solved using numbers but it takes too much of time generally and getting across algebra solution is best if the method strikes early which in my case is a 50-50 chance
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Re: If n is a positive integer, is the value of x - y at least  [#permalink]

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23 Aug 2019, 02:11
chondro48 wrote:
(1) $$x-y=5^{n-1}-3^{n+1}=3*(5^n-3^n) - \frac{14}{5}*5^n$$. Since $$\frac{14}{5}*5^n > 0$$, then it must be true that $$x-y < 3*(5^n-3^n)$$.
SUFFICIENT

(2) $$x-y=5^{n+1}-3^{n+1}=3*(5^n-3^n) + 2*5^n$$. Since $$2*5^n > 0$$, then it must be true that $$x-y > 3*(5^n-3^n)$$.
SUFFICIENT

+1 kudo is appreciated =)

Hi chondro48,
Appreciate your adaptation, but you have been a member of GMAT club for about two years now, you should know that in an actual GMAT question , two statements should never contradict each other. In your first statement you have $$x-y < 3(5^n−3^n)$$ and in your second statement you have $$x-y > 3(5^n−3^n)$$.

This should not happen in an actual GMAT question !
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Re: If n is a positive integer, is the value of x - y at least  [#permalink]

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Updated on: 24 Aug 2019, 19:32
stne wrote:
chondro48 wrote:
(1) $$x-y=5^{n-1}-3^{n+1}=3*(5^n-3^n) - \frac{14}{5}*5^n$$. Since $$\frac{14}{5}*5^n > 0$$, then it must be true that $$x-y < 3*(5^n-3^n)$$.
SUFFICIENT

(2) $$x-y=5^{n+1}-3^{n+1}=3*(5^n-3^n) + 2*5^n$$. Since $$2*5^n > 0$$, then it must be true that $$x-y > 3*(5^n-3^n)$$.
SUFFICIENT

+1 kudo is appreciated =)

Hi chondro48,
Appreciate your adaptation, but you have been a member of GMAT club for about two years now, you should know that in an actual GMAT question , two statements should never contradict each other. In your first statement you have $$x-y < 3(5^n−3^n)$$ and in your second statement you have $$x-y > 3(5^n−3^n)$$.

This should not happen in an actual GMAT question !

Hi stne,

Thank you for the input. The question has been revised so that the two statements do not contradict each other and the answer key remains the same. After all, the old question is still great for practicing concepts and creativity.

Posted from my mobile device

Originally posted by chondro48 on 23 Aug 2019, 02:52.
Last edited by chondro48 on 24 Aug 2019, 19:32, edited 2 times in total.
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Re: If n is a positive integer, is the value of x - y at least  [#permalink]

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24 Aug 2019, 07:38
chondro48 wrote:
If $$n$$ is a positive integer, is the value of $$x - y$$ at least three times the value of $$5^n−3^n$$ ?

(1) $$x=5^{n-1}$$ and $$y=3^{n+1}$$

(2) $$x=5^{n+1}$$ and $$y=3^{n+1}$$

+1 kudo is appreciated.

So if I take value of n as 1, I get 5-3 as 2. So effectively I am checking if x-y >=6.

A. x=1 y=9. x-y becomes a negative value. I mean sure the question does not say x-y is positive but it can't be greater than 6 for sure. And n can be 1 because it is a positive number which means 1 is not sufficient.

Am I doing something wrong?

Thanks for the tag!
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Re: If n is a positive integer, is the value of x - y at least  [#permalink]

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Updated on: 24 Aug 2019, 12:55
TheNightKing wrote:
chondro48 wrote:
If $$n$$ is a positive integer, is the value of $$x - y$$ at least three times the value of $$5^n−3^n$$ ?

(1) $$x=5^{n-1}$$ and $$y=3^{n+1}$$

(2) $$x=5^{n+1}$$ and $$y=3^{n+1}$$

So if I take value of n as 1, I get 5-3 as 2. So effectively I am checking if x-y >=6.

A. x=1 y=9. x-y becomes a negative value. I mean sure the question does not say x-y is positive but it can't be greater than 6 for sure. And n can be 1 because it is a positive number which means 1 is not sufficient.

Am I doing something wrong?

Thanks for the tag!

Hi TheNightKing,

No, you are not that wrong. If $$n = 1$$, then $$x - y = 1 - 9 = -8$$, certainly less than three times $$(5^1 - 3^1) = 6$$. If you try any other value of positive integer n (n = 2, 3, 4, etc), you will find that the value of $$x - y$$ is ALWAYS less than three times the value of $$5^n−3^n$$. Thus, statement (1) is indeed sufficient.

Similar scheme (pick a number) can also be applied for statement (2).

Btw. The question actually doesn't limit that n has to be 1.

Originally posted by chondro48 on 24 Aug 2019, 07:52.
Last edited by chondro48 on 24 Aug 2019, 12:55, edited 3 times in total.
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Re: If n is a positive integer, is the value of x - y at least  [#permalink]

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24 Aug 2019, 07:55
chondro48 wrote:
TheNightKing wrote:
chondro48 wrote:
If $$n$$ is a positive integer, is the value of $$x - y$$ at least three times the value of $$5^n−3^n$$ ?

(1) $$x=5^{n-1}$$ and $$y=3^{n+1}$$

(2) $$x=5^{n+1}$$ and $$y=3^{n+1}$$

So if I take value of n as 1, I get 5-3 as 2. So effectively I am checking if x-y >=6.

A. x=1 y=9. x-y becomes a negative value. I mean sure the question does not say x-y is positive but it can't be greater than 6 for sure. And n can be 1 because it is a positive number which means 1 is not sufficient.

Am I doing something wrong?

Thanks for the tag!

No, you are not wrong. When n=1, it is certain that x-y=1-9=-8 is less than three times (5^1 - 3^1)=6. This applies to any value of n (please try n=2 or n=3). Therefore, statement (1) is deemed sufficient.

Btw. The question actually just asks whether $$x - y$$ is at least three times the value of $$5^n−3^n$$, not what is the value of n.

generously give kudo to the question. Thanks buddy:)

Posted from my mobile device

As far as I know that is not how DS works, If it doesn't work for n=1 but works for n>1 and the question is saying n is positive number and not n is positive number greater than 1 that means the statement is not sufficient.
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Re: If n is a positive integer, is the value of x - y at least  [#permalink]

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24 Aug 2019, 10:28
chondro48 wrote:
If $$n$$ is a positive integer, is the value of $$x - y$$ at least three times the value of $$5^n−3^n$$ ?

(1) $$x=5^{n-1}$$ and $$y=3^{n+1}$$

(2) $$x=5^{n+1}$$ and $$y=3^{n+1}$$

+1 kudo is appreciated.

about what level is this question? 600+?
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Re: If n is a positive integer, is the value of x - y at least  [#permalink]

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24 Aug 2019, 12:56
DiogoNunes wrote:
chondro48 wrote:
If $$n$$ is a positive integer, is the value of $$x - y$$ at least three times the value of $$5^n−3^n$$ ?

(1) $$x=5^{n-1}$$ and $$y=3^{n+1}$$

(2) $$x=5^{n+1}$$ and $$y=3^{n+1}$$

+1 kudo is appreciated.

about what level is this question? 600+?

Just medium, 600-650 level.
Re: If n is a positive integer, is the value of x - y at least   [#permalink] 24 Aug 2019, 12:56
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