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Is 5^k less than 1,000?

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Is 5^k less than 1,000? [#permalink] New post 07 Dec 2010, 17:19
00:00
A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

65% (02:25) correct 35% (01:13) wrong based on 67 sessions
Is 5^k less than 1,000?

(1) 5^(k+1) > 3,000

(2) 5^(k-1) = (5^k) - 500

OPEN DISCUSSION OF THIS QUESTION IS HERE: is-5-k-less-than-1-000-1-5-k-128055.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 23 May 2014, 01:02, edited 2 times in total.
Renamed the topic, edited the question and added the OA.
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Re: Variable in Exponent DS problem [#permalink] New post 07 Dec 2010, 17:50
Expert's post
tonebeeze wrote:
I would appreciate it if someone could walk me through this problem. Thanks

Is 5^k less than 1000?

1. 5^k+1 > 3000
2. 5^k-1 = 5^k -500


I am assuming the question is:

Is 5^k less than 1000?

1. 5^{k+1} > 3000
2. 5^{k-1} = 5^k -500

5^4 = 625 and 5^5 = 3125 (even if you do not know this, it is fine. You don't need to calculate. Just observe that 625*5 will be greater than 3000)

Statement 1: 5^{k+1} > 3000
This means k + 1 is greater than 4 so k is greater than 3 (It doesnt mean that k + 1 is at least 5 because the question doesn't say that k is an integer. k + 1 could be 4.999 making k = 3.999) Since k can take values less than 4 and more than 4, 5^k could be less than 1000 or more than 1000. Not sufficient.

Statement 2: 5^{k-1} = 5^k -500
Re-arrange: 500 = 5^k -5^{k-1}
5^3 *4 = 5^{k-1}(5 - 1)
Hence k - 1 = 3 and k = 4
So 5^k = 625 which is less than 1000. Answer is 'Yes'. Sufficient.

Answer (B)
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Re: Variable in Exponent DS problem [#permalink] New post 07 Dec 2010, 18:18
Great point that (K+1) in stmt 1 does not mean more than 4 or at least 5, we forget non-integers when considering DS questions. Thanks.
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Re: Variable in Exponent DS problem [#permalink] New post 02 Jan 2011, 19:03
Can you go over how you went from 500 = 5^k -5^{k-1} to
5^3 *4 = 5^{k-1}(5 - 1)?

Thanks so much in advance!
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Re: Variable in Exponent DS problem [#permalink] New post 02 Jan 2011, 19:30
m990540 wrote:
Can you go over how you went from 500 = 5^k -5^{k-1} to
5^3 *4 = 5^{k-1}(5 - 1)?

Thanks so much in advance!



Please refer to this link. It will help you easily post mathematic symbols in your posts.

writing-mathematical-symbols-in-posts-72468.html

Regarding the problem:
500 = 5^k - 5^{k-1}

500 can be simplified into 125 x 4, which is equal to 5^3 x 4

Step 1: Recognize that 5^k is a common factor in both 5^k and 5^{k -1}. Proceed to factor out 5^k

Step 2: 5^k (1 - 5^{-1}) =500 ---> 5^k (1 - \frac {1}{5}) = 500 ---> 5k (\frac {4}{5}) = 500 --->5^k = \frac {5}{4} (500) ---> 5^k = 625

Hope this helps
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Re: Variable in Exponent DS problem [#permalink] New post 03 Jan 2011, 07:29
Expert's post
m990540 wrote:
Can you go over how you went from 500 = 5^k -5^{k-1} to
5^3 *4 = 5^{k-1}(5 - 1)?

Thanks so much in advance!


Left hand side:
500 = 5*100 = 5*25*4 = 5^3*4 (Since your concern is the power of 5, separate 5s from the rest)

Right hand side:
5^k -5^{k-1} = 5^{k - 1} ( 5 - 1)(Take 5^{k - 1} common. e.g. if you have 5^4 - 5^3, you can take 5^3 common and you will be left with (5 - 1))
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Re: Variable in Exponent DS problem [#permalink] New post 03 Jan 2011, 08:14
5^k+1 > 3000
5. 5^k > 3000
5^k > 600 (This is not sufficient to tell that it is less than 1000)
5^k-1 = 5^k -500
5^k / 5 = 5^k – 500
5^k = 5^k+1 – 2500
5^k (5 -1) = 2500
5^k = 2500 / 4 = 625 (this is sufficient)
Answer is (B)
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Re: Variable in Exponent DS problem [#permalink] New post 23 Nov 2011, 10:51
St1: 5^k >600 ----insufficient as we don't know if 5^k <1000

St2: 5 ^ {k-1} = 5^K - 500
500 = 5^k (1- \frac{1}{5})
500 (\frac{5}{4})= 5^k ----sufficient as we can determine if 5^k <1000


hence B!
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Re: Variable in Exponent DS problem [#permalink] New post 22 May 2014, 22:11
VeritasPrepKarishma wrote:
tonebeeze wrote:
I would appreciate it if someone could walk me through this problem. Thanks

Is 5^k less than 1000?

1. 5^k+1 > 3000
2. 5^k-1 = 5^k -500


I am assuming the question is:

Is 5^k less than 1000?

1. 5^{k+1} > 3000
2. 5^{k-1} = 5^k -500

5^4 = 625 and 5^5 = 3125 (even if you do not know this, it is fine. You don't need to calculate. Just observe that 625*5 will be greater than 3000)

Statement 1: 5^{k+1} > 3000
This means k + 1 is greater than 4 so k is greater than 3 (It doesnt mean that k + 1 is at least 5 because the question doesn't say that k is an integer. k + 1 could be 4.999 making k = 3.999) Since k can take values less than 4 and more than 4, 5^k could be less than 1000 or more than 1000. Not sufficient.

Statement 2: 5^{k-1} = 5^k -500
Re-arrange: 500 = 5^k -5^{k-1}
5^3 *4 = 5^{k-1}(5 - 1)
Hence k - 1 = 3 and k = 4
So 5^k = 625 which is less than 1000. Answer is 'Yes'. Sufficient.

Answer (B)


I can't get why are we talking about decimals in the first statement. I don't know if I'm doing something wrong here...

I did it like this
5^{k+1} > 3000
k+1 \geq 5
k \geq 4

k= 4 then 5^4 = 625
k=5 then 5^5 = 3000 (approx)
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Re: Variable in Exponent DS problem [#permalink] New post 23 May 2014, 01:12
Expert's post
b2bt wrote:
VeritasPrepKarishma wrote:
tonebeeze wrote:
I would appreciate it if someone could walk me through this problem. Thanks

Is 5^k less than 1000?

1. 5^k+1 > 3000
2. 5^k-1 = 5^k -500


I am assuming the question is:

Is 5^k less than 1000?

1. 5^{k+1} > 3000
2. 5^{k-1} = 5^k -500

5^4 = 625 and 5^5 = 3125 (even if you do not know this, it is fine. You don't need to calculate. Just observe that 625*5 will be greater than 3000)

Statement 1: 5^{k+1} > 3000
This means k + 1 is greater than 4 so k is greater than 3 (It doesnt mean that k + 1 is at least 5 because the question doesn't say that k is an integer. k + 1 could be 4.999 making k = 3.999) Since k can take values less than 4 and more than 4, 5^k could be less than 1000 or more than 1000. Not sufficient.

Statement 2: 5^{k-1} = 5^k -500
Re-arrange: 500 = 5^k -5^{k-1}
5^3 *4 = 5^{k-1}(5 - 1)
Hence k - 1 = 3 and k = 4
So 5^k = 625 which is less than 1000. Answer is 'Yes'. Sufficient.

Answer (B)


I can't get why are we talking about decimals in the first statement. I don't know if I'm doing something wrong here...

I did it like this
5^{k+1} > 3000
k+1 \geq 5
k \geq 4

k= 4 then 5^4 = 625
k=5 then 5^5 = 3000 (approx)


5^{k+1} > 3000 --> 5*5^{k} > 3000 --> 5^{k} > 600. Now, we are NOT told that k is an integer, thus we cannot say that k\geq{4}. For example, 5^{3.99}\approx{615}, thus k could be 3.99.

Is 5^k less than 1,000?

Is 5^k<1,000?

(1) 5^(k+1) > 3,000 --> 5^k>600 --> if k=4 then the answer is YES: since 600<(5^4=625)<1,000 but if k=10, for example, then the answer is NO. Not sufficient.

(2) 5^(k-1) = (5^k) - 500 --> we can solve for k and get the single numerical value of it, hence this statement is sufficient. Just to illustrate: 5^k-5^{k-1}=500 --> factor out 5^{k-1}: 5^{k-1}(5-1)=500 --> 5^{k-1}=125 --> k-1=3 --> k=4. Sufficient.

Answer: B.

Hope it's clear.

OPEN DISCUSSION OF THIS QUESTION IS HERE: is-5-k-less-than-1-000-1-5-k-128055.html
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Re: Variable in Exponent DS problem   [#permalink] 23 May 2014, 01:12
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