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# If n is a positive integer, is (1/2)^n > 0.125 ?

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Math Expert
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If n is a positive integer, is (1/2)^n > 0.125 ?  [#permalink]

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21 Aug 2019, 08:34
00:00

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45% (medium)

Question Stats:

65% (01:32) correct 35% (01:27) wrong based on 68 sessions

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Competition Mode Question

If n is a positive integer, is $$(\frac{1}{2})^n > 0.125$$ ?

(1) $$n > 3$$

(2) $$(\frac{1}{2})^{n - 1} < 0.25$$

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Re: If n is a positive integer, is (1/2)^n > 0.125 ?  [#permalink]

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Updated on: 21 Aug 2019, 08:55
1
Quote:
If n is a positive integer, is $$(\frac{1}{2})^n>0.125?$$

(1) n>3

(2) $$(\frac{1}{2})^n−1<0.25$$

Given: n is a positive integer

Asked: Is $$(\frac{1}{2})^n>0.125?$$

$$0.125 = (\frac{1}{2})^3$$
Is $$(\frac{1}{2})^n>0.125$$
Mean Is n<3?

(1) n>3
If n>3 => $$(\frac{1}{2})^n<0.125$$
SUFFICIENT

(2) $$(\frac{1}{2})^{n−1}<0.25$$
n-1>2
n>3
If n>3 => $$(\frac{1}{2})^n<0.125$$
SUFFICIENT

IMO D
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Originally posted by Kinshook on 21 Aug 2019, 08:46.
Last edited by Kinshook on 21 Aug 2019, 08:55, edited 3 times in total.
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Re: If n is a positive integer, is (1/2)^n > 0.125 ?  [#permalink]

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21 Aug 2019, 08:51
1
IMO-D

n is a positive integer
Check : 1/2 ^ n> 0.125 ?

0.125 = 1/2 ^ 3
So, Check (1/2)^n > (1/2)^3

(1) n>3
If n>3, (1/2)^n < (1/2)^3
Sufficient

(2) 1/2 ^ (n−1) <0.25
1/2 ^ (n−1) < (1/2)^2
n-1 > 2
n>3
If n>3, (1/2)^n < (1/2)^3
Sufficient
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If n is a positive integer, is (1/2)^n > 0.125 ?  [#permalink]

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Updated on: 22 Aug 2019, 09:00
1
n = positive integer

We need to find whether $$(\frac{1}{2})^n > 0.125$$?
=$$(\frac{1}{2})^n > (\frac{125}{1000})$$?
=$$(\frac{1}{2})^n > (\frac{1}{8})$$?
=$$(\frac{1}{2})^n > (\frac{1}{2})^3$$?

(1) n>3 Any number greater than 3 would always give NO.
Example:
when n=4 => $$(\frac{1}{2})^4 > (\frac{1}{2})^3$$? NO
when n=5 => $$(\frac{1}{2})^5 > (\frac{1}{2})^3$$? NO

Hence n always has to be less than 3, i.e, n<3

SUFFICIENT!

(2) $$(\frac{1}{2})^{(n-1)} < 0.25$$
$$(\frac{1}{2})^{(n-1)} < (\frac{25}{100})$$
$$(\frac{1}{2})^{(n-1)} < (\frac{1}{2})^2$$
n-1 > 2 (sign changes here)
n>3.............same as statement 1, always no!
SUFFICIENT

IMO OPTION D
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Originally posted by EncounterGMAT on 21 Aug 2019, 09:03.
Last edited by EncounterGMAT on 22 Aug 2019, 09:00, edited 2 times in total.
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Re: If n is a positive integer, is (1/2)^n > 0.125 ?  [#permalink]

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21 Aug 2019, 09:04
n should be less than 3 to make sure $$(1/2)^n > 0.125$$ as (1/2)^n > (1/2)^3 => n < 3 (Fractional value decreases as power increases)

1) This is sufficient as n > 3, so answer is NO

2) With the same logic, n -1 > 2 => n > 3 , sufficient as it would result in answer NO

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Re: If n is a positive integer, is (1/2)^n > 0.125 ?  [#permalink]

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21 Aug 2019, 09:22
If n is a positive integer, is (12)n>0.125(12)n>0.125 ?

(1) n>3

(2) (12)n−1<0.25

simplify given expression
we get
(1/2)^n>(1/2)^3
only possible when n=1,2
#1
n>3
sufficient
#2
(12)n−1<0.25
solve we get (1/2)^n-1<(1/2)^2
possible when n>3
sufficient
IMO D
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Re: If n is a positive integer, is (1/2)^n > 0.125 ?  [#permalink]

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21 Aug 2019, 09:38
statement 1 gives

n>3 so 1/2^3 = .125

so if n >3 then it will be lesser than .125 which is clear no so sufficient

statement 2

1/2 ^(n-1)

1/2^n/1/2^1

multiply both side by 1/2 gives

1/2 ^n > .125 which is nothing but question so not sufficient

hence ans is A
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Re: If n is a positive integer, is (1/2)^n > 0.125 ?  [#permalink]

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21 Aug 2019, 09:48
If n is a positive integer, is $$(\frac{1}{2})^n > 0.125$$ ?

(1) n > 3

(2) $$\frac{1}{2}^(n−1)$$ < 0.25

$$(\frac{1}{2})^n > 0.125$$
 $$(\frac{1}{2})^n > (\frac{1}{8})$$ [Also $$0.125 = (0.5)^3$$]
 $$(\frac{1}{2})^n > (\frac{1}{2})^3$$
 n > 3 [Since n >0]

Statement 1) n > 3

Yes n > 3

SUFFICIENT

Statement 2) $$\frac{1}{2}^(n−1)$$ < 0.25
 $$\frac{1}{2}^(n−1)$$ < $$(0.5)^2$$
 n - 1 < 2 [Since n >0]
 n < 3
No, n > 3

SUFFICIENT

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Re: If n is a positive integer, is (1/2)^n > 0.125 ?  [#permalink]

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21 Aug 2019, 11:07
If n is a positive integer, is $$(\frac{1}{2})^{n}$$ > 0.125 ?

$$(\frac{1}{2})^{n}$$ > $$\frac{1}{8}$$

$$(\frac{1}{2})^{n}$$ > $$(\frac{1}{2})^{3}$$

is n < 3 ???

Statement1: n>3
--> Always NO.
Sufficient

Statement2: $$(\frac{1}{2})^{n−1}$$ < 0.25

$$(\frac{1}{2})^{n-1}$$ < $$(\frac{1}{2})^2$$
n-1 > 2
n >3
Always YES
Sufficient

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Re: If n is a positive integer, is (1/2)^n > 0.125 ?  [#permalink]

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21 Aug 2019, 11:38
1
Given: n >=1
Analysis: (1/2)^n > 0.125
(1/2)^n > 1/8 => n>3

St.1 : n>3 Sufficient
St.2 : (1/2)^n * 2 < 1/4
(1/2)^n < 1/8
n < 3 => Sufficient.

D ans.
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Re: If n is a positive integer, is (1/2)^n > 0.125 ?  [#permalink]

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21 Aug 2019, 11:43
1st Statement
n>3 , (1/2)^4 = 0.0625
Insuff. for the question being.

2nd Statement
(1/2)^n-1 <0.25
If n= 1,2,3 then statement false.
Insuff. for the question being

But (1) + (2)
n>3 and (1/2)^n-1 <0.25
n=4 , the answer comes to be 0.125<0.25
Hence together Sufficient.

IMO C
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Re: If n is a positive integer, is (1/2)^n > 0.125 ?  [#permalink]

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21 Aug 2019, 12:00
We know from the question stem that n is a positive integer. We are to find if (0.5)^n > (0.5)^3. This means we are simply asked if n>3.

Statement 1 says n>3. Clearly it is sufficient since we can answer that n>3 or that for all integers values n={4,5,5,...}, (0.5)^n is not greater than 0.125.

Statement 2: (0.5)^(n-1) < (0.5)^2
This means that n-1 < 2 hence n<3.
And for all positive values of n<3 (1,2), (0.5)^n > 0.125.
Hence we can answer yes to the question posed. Statement 2 is also sufficient on its own.

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Re: If n is a positive integer, is (1/2)^n > 0.125 ?  [#permalink]

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21 Aug 2019, 12:38
Quote:
If n is a positive integer, is (1/2)^n>0.125 ?

(1) n>3
(2) (1/2)^(n−1)<0.25

$$(1/2)^n>1/8…2^{(-n)}>2^{(-3)}…-n>-3…n<3$$ so, find if $$n<3$$

(1) n>3: sufic.
(2) (1/2)^(n−1)<0.25: $$(1/2)^{(n-1)}>1/4…2^{(-(n-1))}>2^{(-2)}…-(n-1)>-2…-n>-3…n<3$$; sufic.

Re: If n is a positive integer, is (1/2)^n > 0.125 ?   [#permalink] 21 Aug 2019, 12:38
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