Check GMAT Club Decision Tracker for the Latest School Decision Releases https://gmatclub.com/AppTrack

 It is currently 25 May 2017, 14:48

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

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

Author Message
TAGS:

### Hide Tags

Manager
Joined: 02 Dec 2012
Posts: 178
Followers: 6

Kudos [?]: 2695 [1] , given: 0

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

### Show Tags

18 Dec 2012, 09:00
1
KUDOS
35
This post was
BOOKMARKED
00:00

Difficulty:

55% (hard)

Question Stats:

63% (02:36) correct 37% (01:45) wrong based on 1009 sessions

### HideShow timer Statistics

If n is a positive integer, is the value of b - a at least twice the value of 3^n - 2^n?

(1) a= 2^(n+1) and b= 3^(n+1)
(2) n = 3
[Reveal] Spoiler: OA
Math Expert
Joined: 02 Sep 2009
Posts: 38881
Followers: 7733

Kudos [?]: 106125 [0], given: 11607

Re: If n is a positive integer, is the value of b - a at least [#permalink]

### Show Tags

18 Dec 2012, 09:03
Expert's post
7
This post was
BOOKMARKED
If n is a positive integer, is the value of b - a at least twice the value of 3^n - 2^n?

Question: is $$b-a\geq{2(3^n - 2^n)}$$?

(1) a= 2^(n+1) and b= 3^(n+1). The question becomes: is $$3^{n+1}-2^{n+1}\geq{2*(3^n - 2^n)}$$? --> is $$3*3^{n}-2*2^{n}\geq{2*3^n-2*2^{n}$$? --> is $$3^{n}\geq{0}$$? 3^n is always more than zero, so this statement is sufficient.

(2) n = 3. Clearly insufficient.

_________________
Intern
Joined: 02 Jul 2013
Posts: 19
Schools: LBS MIF '15
Followers: 0

Kudos [?]: 135 [0], given: 16

Re: If n is a positive integer, is the value of b - a at least [#permalink]

### Show Tags

29 Sep 2013, 02:59
Bunuel wrote:
If n is a positive integer, is the value of b - a at least twice the value of 3^n - 2^n?

Question: is $$b-a\geq{2(3^n - 2^n)}$$?

(1) a= 2^(n+1) and b= 3^(n+1). The question becomes: is $$3^{n+1}-2^{n+1}\geq{2*(3^n - 2^n)}$$? --> is $$3*3^{n}-2*2^{n}\geq{2*3^n-2*2^{n}$$? --> is $$3^{n}\geq{0}$$? 3^n is always more than zero, so this statement is sufficient.

(2) n = 3. Clearly insufficient.

For statement (1), I understand how you arrived at $$3*3^{n}-2*2^{n}\geq{2*3^n-2*2^{n}$$ but how does this equation tell you whether b-a is at least twice the value of 3^n-2^n?

For example, if i sub in n=1 into the equation, it is true that b-a is at least twice the value (b-a= 5 and 3^n-2^n = 1 then 1*2). However, if i sub in n=2, the equation no longer is true as b-a then becomes 19 and the right side of the equation then becomes 10, which is not at least twice the value.

What am i missing here?
Math Expert
Joined: 02 Sep 2009
Posts: 38881
Followers: 7733

Kudos [?]: 106125 [0], given: 11607

Re: If n is a positive integer, is the value of b - a at least [#permalink]

### Show Tags

29 Sep 2013, 13:01
bulletpoint wrote:
Bunuel wrote:
If n is a positive integer, is the value of b - a at least twice the value of 3^n - 2^n?

Question: is $$b-a\geq{2(3^n - 2^n)}$$?

(1) a= 2^(n+1) and b= 3^(n+1). The question becomes: is $$3^{n+1}-2^{n+1}\geq{2*(3^n - 2^n)}$$? --> is $$3*3^{n}-2*2^{n}\geq{2*3^n-2*2^{n}$$? --> is $$3^{n}\geq{0}$$? 3^n is always more than zero, so this statement is sufficient.

(2) n = 3. Clearly insufficient.

For statement (1), I understand how you arrived at $$3*3^{n}-2*2^{n}\geq{2*3^n-2*2^{n}$$ but how does this equation tell you whether b-a is at least twice the value of 3^n-2^n?

For example, if i sub in n=1 into the equation, it is true that b-a is at least twice the value (b-a= 5 and 3^n-2^n = 1 then 1*2). However, if i sub in n=2, the equation no longer is true as b-a then becomes 19 and the right side of the equation then becomes 10, which is not at least twice the value.

What am i missing here?

For the first statement after some manipulation the question becomes: is $$3^{n}\geq{0}$$? Irrespective of the actual value of n, 3^n will always be greater than zero, thus the answer to the question is YES.
_________________
Manager
Joined: 29 Sep 2013
Posts: 53
Followers: 0

Kudos [?]: 36 [2] , given: 48

Re: If n is a positive integer, is the value of b - a at least [#permalink]

### Show Tags

30 Sep 2013, 18:49
2
KUDOS
2
This post was
BOOKMARKED
If n is a positive integer, is the value of b - a at least twice the value of 3^n - 2^n?

(1) a= 2^(n+1) and b= 3^(n+1)
(2) n = 3

The Question asks: b - a > 2*(3^n - 2^n)

The answer should be a definitive yes/no

Statement 1:

a= 2^(n+1) and b= 3^(n+1)

Lets take the smallest possible value of a Positive Integer i.e 1 and put it in for "n"

3^(1+1) - 2^ (1+1) > 2*(3^1 - 2^1)
3^2 - 2^2 > 2* (3 - 2)
9 - 4 > 2 *1

Lets test another number (just to be on the Safe Side), lets test n=2

3^(2+1) - 2^ (2+1) > 2*(3^2 - 2^2)
3^3 - 2^3 > 2* (9 - 4)
27 - 8 > 2* 5

Thus, Sufficient.

Statement 2:

n=3

Let's Put it in the in-equality b - a > 2*(3^n - 2^n)

b - a> 2*(3^3 - 2^3)
b - a> 2*(27 - 8)
b - a> 2*(19)
b - a> 38

If, b=100 and a=10, than definitive answer
If, b= 2 and a= 10, than definitive answer
But since it doesn't provide any value for either "a" or "b"

Thus, Not Sufficient

[Reveal] Spoiler:
A
Intern
Joined: 18 Dec 2013
Posts: 37
Location: United States
Followers: 0

Kudos [?]: 11 [0], given: 19

Re: If n is a positive integer, is the value of b - a at least [#permalink]

### Show Tags

20 Jul 2014, 00:12
Do you think that it's better to approach this problem by plugging in numbers for n (as long as you're sure you can do it fast for cases n=1 and n=2), or to do the problem algebraically (which you know will take ~45-90seconds to crunch)?

Would be particularly interested in getting Bunuel's take...Thanks!
Math Expert
Joined: 02 Sep 2009
Posts: 38881
Followers: 7733

Kudos [?]: 106125 [0], given: 11607

Re: If n is a positive integer, is the value of b - a at least [#permalink]

### Show Tags

20 Jul 2014, 06:06
warriorsquared wrote:
Do you think that it's better to approach this problem by plugging in numbers for n (as long as you're sure you can do it fast for cases n=1 and n=2), or to do the problem algebraically (which you know will take ~45-90seconds to crunch)?

Would be particularly interested in getting Bunuel's take...Thanks!

You should take the approach which fits you the best.
_________________
Intern
Status: Preparing for GMAT
Joined: 10 Dec 2013
Posts: 20
Location: India
GMAT 1: 530 Q46 V18
WE: Other (Entertainment and Sports)
Followers: 0

Kudos [?]: 7 [0], given: 61

Re: If n is a positive integer, is the value of b - a at least [#permalink]

### Show Tags

20 Jul 2014, 09:17
Can someone please show me the simplification as to how the equation becomes whether 3^n > 0?
Math Expert
Joined: 02 Sep 2009
Posts: 38881
Followers: 7733

Kudos [?]: 106125 [2] , given: 11607

Re: If n is a positive integer, is the value of b - a at least [#permalink]

### Show Tags

20 Jul 2014, 09:43
2
KUDOS
Expert's post
5
This post was
BOOKMARKED
suhaschan wrote:
Can someone please show me the simplification as to how the equation becomes whether 3^n > 0?

$$3^{n+1}-2^{n+1}\geq{2*(3^n - 2^n)}$$;

$$3*3^{n}-2*2^{n}\geq{2*3^n-2*2^{n}$$;

Cancel $$-2*2^{n}$$: $$3*3^{n}\geq{2*3^n}$$;

Subtract 2*3^n from both sides: $$3*3^{n}-2*3^n\geq{0}$$;

$$3^{n}\geq{0}$$

Hope it's clear.
_________________
Intern
Status: Preparing for GMAT
Joined: 10 Dec 2013
Posts: 20
Location: India
GMAT 1: 530 Q46 V18
WE: Other (Entertainment and Sports)
Followers: 0

Kudos [?]: 7 [0], given: 61

Re: If n is a positive integer, is the value of b - a at least [#permalink]

### Show Tags

21 Jul 2014, 05:50
Thanks a lot Bunuel. The explanation couldn't have been more lucid.
Current Student
Joined: 13 Feb 2011
Posts: 104
Followers: 0

Kudos [?]: 38 [1] , given: 3370

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

### Show Tags

03 Aug 2014, 10:09
1
KUDOS
Bunuel wrote:
suhaschan wrote:
Can someone please show me the simplification as to how the equation becomes whether 3^n > 0?

$$3^{n+1}-2^{n+1}\geq{2*(3^n - 2^n)}$$;

$$3*3^{n}-2*2^{n}\geq{2*3^n-2*2^{n}$$;

Cancel $$-2*2^{n}$$: $$3*3^{n}\geq{2*3^n}$$;

Subtract 2*3^n from both sides: $$3*3^{n}-2*3^n\geq{0}$$;

$$3^{n}\geq{0}$$

Hope it's clear.

Hi Bunuel,
In the end instead of subtracting $$2*3^n$$ from both sides, can we cancel $$3^n$$ from both sides (as it's always positive) and reach $$3>2$$ making the statement sufficient? Is that also a correct approach?
Thanks.
Math Expert
Joined: 02 Sep 2009
Posts: 38881
Followers: 7733

Kudos [?]: 106125 [2] , given: 11607

Re: If n is a positive integer, is the value of b - a at least [#permalink]

### Show Tags

12 Aug 2014, 02:55
2
KUDOS
Expert's post
Dienekes wrote:
Bunuel wrote:
suhaschan wrote:
Can someone please show me the simplification as to how the equation becomes whether 3^n > 0?

$$3^{n+1}-2^{n+1}\geq{2*(3^n - 2^n)}$$;

$$3*3^{n}-2*2^{n}\geq{2*3^n-2*2^{n}$$;

Cancel $$-2*2^{n}$$: $$3*3^{n}\geq{2*3^n}$$;

Subtract 2*3^n from both sides: $$3*3^{n}-2*3^n\geq{0}$$;

$$3^{n}\geq{0}$$

Hope it's clear.

Hi Bunuel,
In the end instead of subtracting $$2*3^n$$ from both sides, can we cancel $$3^n$$ from both sides (as it's always positive) and reach $$3>2$$ making the statement sufficient? Is that also a correct approach?
Thanks.

Yes, this also would be a correct way of solving.
_________________
Manager
Joined: 26 Feb 2015
Posts: 127
Followers: 0

Kudos [?]: 15 [0], given: 43

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

### Show Tags

04 May 2015, 06:45
Bunuel wrote:
suhaschan wrote:
Can someone please show me the simplification as to how the equation becomes whether 3^n > 0?

$$3^{n+1}-2^{n+1}\geq{2*(3^n - 2^n)}$$;

$$3*3^{n}-2*2^{n}\geq{2*3^n-2*2^{n}$$;

Cancel $$-2*2^{n}$$: $$3*3^{n}\geq{2*3^n}$$;

Subtract 2*3^n from both sides: $$3*3^{n}-2*3^n\geq{0}$$;

$$3^{n}\geq{0}$$

Hope it's clear.

You could stop at the part I have boldfaced to realize that the statement is sufficient, no? I mean $$3*3^{n}-2*2^{n}\geq{2*3^n-2*2^{n}$$; clearly shows that it it IS bigger. (Given that n is a positive integer)
Intern
Joined: 14 Oct 2015
Posts: 37
GMAT 1: 640 Q45 V33
Followers: 0

Kudos [?]: 16 [0], given: 0

Re: If n is a positive integer, is the value of b - a at least [#permalink]

### Show Tags

22 Oct 2015, 14:20
bulletpoint wrote:
Bunuel wrote:
If n is a positive integer, is the value of b - a at least twice the value of 3^n - 2^n?

Question: is $$b-a\geq{2(3^n - 2^n)}$$?

(1) a= 2^(n+1) and b= 3^(n+1). The question becomes: is $$3^{n+1}-2^{n+1}\geq{2*(3^n - 2^n)}$$? --> is $$3*3^{n}-2*2^{n}\geq{2*3^n-2*2^{n}$$? --> is $$3^{n}\geq{0}$$? 3^n is always more than zero, so this statement is sufficient.

(2) n = 3. Clearly insufficient.

For statement (1), I understand how you arrived at $$3*3^{n}-2*2^{n}\geq{2*3^n-2*2^{n}$$ but how does this equation tell you whether b-a is at least twice the value of 3^n-2^n?

For example, if i sub in n=1 into the equation, it is true that b-a is at least twice the value (b-a= 5 and 3^n-2^n = 1 then 1*2). However, if i sub in n=2, the equation no longer is true as b-a then becomes 19 and the right side of the equation then becomes 10, which is not at least twice the value.

What am i missing here?

How did you get that the right side of the equation is 10?? 3^2 = 9 and 2^2 = 4 which means 5. 5 X 2 = 10. 19 > 10.
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15444
Followers: 649

Kudos [?]: 209 [0], given: 0

Re: If n is a positive integer, is the value of b - a at least [#permalink]

### Show Tags

11 Jan 2017, 11:21
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Intern
Joined: 18 Jan 2017
Posts: 38
Followers: 0

Kudos [?]: 2 [0], given: 4

Re: If n is a positive integer, is the value of b - a at least [#permalink]

### Show Tags

20 Jan 2017, 03:56
From (1), we have to find whether:

b - a > 2*[3^n - 2^n] ?

Substituting a and b,

3^(n+1) - 2^(n+1) > 2*[3^n - 2^n] ?
3^(n+1) - 2^(n+1) > 2*3^n - 2^(n+1) ?

2^(n+1) gets cancelled on both sides.

3^(n+1) > 2*3^n?
3*3^n > 2*3^n?

3^n gets cancelled on both sides.

3 > 2?

Yes.
Re: If n is a positive integer, is the value of b - a at least   [#permalink] 20 Jan 2017, 03:56
Similar topics Replies Last post
Similar
Topics:
3 IF X and Y are positive integers then what is the least value of their 3 22 Apr 2016, 17:23
11 What is the value of the positive integer n ? 9 24 Feb 2017, 04:36
2 What is the value of positive integer n? 10 15 Oct 2016, 08:35
9 If n is a positive integer, is the value of b - a at least 4 28 Jul 2015, 23:06
4 If n is a positive integer, is n divisible by at least six positive in 4 25 Jan 2017, 13:03
Display posts from previous: Sort by