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Math Expert V
Joined: 02 Sep 2009
Posts: 59573

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Difficulty:   45% (medium)

Question Stats: 66% (02:00) correct 34% (02:09) wrong based on 65 sessions

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The sequence $$a_1$$, $$a_2$$, ... is defined such that $$a_{n}=\frac{a_{n-1}}{n}$$ for all $$n \gt 1$$. How many terms of the sequence are greater than $$\frac{1}{2}$$?

(1) $$a_2 = 5$$

(2) $$a_1 - a_2 = 5$$

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Math Expert V
Joined: 02 Sep 2009
Posts: 59573

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Official Solution:

Basically we have a sequence of numbers which is defined with some formula. For example: $$a_{2}=\frac{a_{1}}{2}$$, $$a_{3}=\frac{a_{2}}{3}$$, $$a_{4}=\frac{a_{3}}{4}$$, ... The question asks: how many numbers from the sequence are greater than $$\frac{1}{2}$$. Notice that if we knew ANY term of the sequence we would be able to get all the other terms/numbers and thus answer the question.

(1) $$a_2=5$$. As discussed above this statement is sufficient as we can write down all the terms. For example: $$a_{2}=\frac{a_{1}}{2}=5$$, which gives $$a_1=10$$. $$a_{3}=\frac{a_{2}}{3}=\frac{5}{3}$$, and so on.

(2) $$a_1-a_2=5$$. Substitute: $$a_1-\frac{a_{1}}{2}=5$$. We can solve for $$a_1$$ and thus will have the same case of knowing one term. Sufficient.

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Intern  Joined: 27 Jun 2013
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GMAT 1: 720 Q50 V37 ### Show Tags

1
There is a mistake in the problem. Since we are asked how many terms are >1/2 we need to know the number of terms in the sequence? Because of this i marked E
Math Expert V
Joined: 02 Sep 2009
Posts: 59573

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susheelshastry wrote:
There is a mistake in the problem. Since we are asked how many terms are >1/2 we need to know the number of terms in the sequence? Because of this i marked E

The question is correct. Please re-read the solution: "Notice that if we knew ANY term of the sequence we would be able to get all the other terms/numbers and thus answer the question."
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Intern  Joined: 31 Dec 2016
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Wouldn't the caveat (for all n > 1) prevent you from solving for a1?
Math Expert V
Joined: 02 Sep 2009
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debglider17 wrote:
Wouldn't the caveat (for all n > 1) prevent you from solving for a1?

n>1 refers to the formula given saying that this formula is true for all n>1.
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Director  G
Joined: 02 Sep 2016
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Bunuel wrote:
susheelshastry wrote:
There is a mistake in the problem. Since we are asked how many terms are >1/2 we need to know the number of terms in the sequence? Because of this i marked E

The question is correct. Please re-read the solution: "Notice that if we knew ANY term of the sequence we would be able to get all the other terms/numbers and thus answer the question."

Hello Bunuel

Even I have the same doubt.

This is a value DS question and to answer this we should know THE EXACT NUMBER OF TERMS IN THIS SEQUENCE greater than 1/2.
But even if we find all the terms of the sequence as you said, we still can't answer how many terms are there EXACTLY.

Thanks.
Math Expert V
Joined: 02 Sep 2009
Posts: 59573

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Shiv2016 wrote:
Bunuel wrote:
susheelshastry wrote:
There is a mistake in the problem. Since we are asked how many terms are >1/2 we need to know the number of terms in the sequence? Because of this i marked E

The question is correct. Please re-read the solution: "Notice that if we knew ANY term of the sequence we would be able to get all the other terms/numbers and thus answer the question."

Hello Bunuel

Even I have the same doubt.

This is a value DS question and to answer this we should know THE EXACT NUMBER OF TERMS IN THIS SEQUENCE greater than 1/2.
But even if we find all the terms of the sequence as you said, we still can't answer how many terms are there EXACTLY.

Thanks.

Try to solve the way proposed in the solution and you'll get EXACT number of terms which are greater than 1/2.
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Intern  B
Joined: 06 Apr 2016
Posts: 22
GMAT 1: 540 Q42 V23 ### Show Tags

Bunuel wrote:
Official Solution:

Basically we have a sequence of numbers which is defined with some formula. For example: $$a_{2}=\frac{a_{1}}{2}$$, $$a_{3}=\frac{a_{2}}{3}$$, $$a_{4}=\frac{a_{3}}{4}$$, ... The question asks: how many numbers from the sequence are greater than $$\frac{1}{2}$$. Notice that if we knew ANY term of the sequence we would be able to get all the other terms/numbers and thus answer the question.

(1) $$a_2=5$$. As discussed above this statement is sufficient as we can write down all the terms. For example: $$a_{2}=\frac{a_{1}}{2}=5$$, which gives $$a_1=10$$. $$a_{3}=\frac{a_{2}}{3}=\frac{5}{3}$$, and so on.

(2) $$a_1-a_2=5$$. Substitute: $$a_1-\frac{a_{1}}{2}=5$$. We can solve for $$a_1$$ and thus will have the same case of knowing one term. Sufficient.

Can anybody reply how can we get the value of n here?
Math Expert V
Joined: 02 Sep 2009
Posts: 59573

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1
saba@4010 wrote:
Bunuel wrote:
Official Solution:

Basically we have a sequence of numbers which is defined with some formula. For example: $$a_{2}=\frac{a_{1}}{2}$$, $$a_{3}=\frac{a_{2}}{3}$$, $$a_{4}=\frac{a_{3}}{4}$$, ... The question asks: how many numbers from the sequence are greater than $$\frac{1}{2}$$. Notice that if we knew ANY term of the sequence we would be able to get all the other terms/numbers and thus answer the question.

(1) $$a_2=5$$. As discussed above this statement is sufficient as we can write down all the terms. For example: $$a_{2}=\frac{a_{1}}{2}=5$$, which gives $$a_1=10$$. $$a_{3}=\frac{a_{2}}{3}=\frac{5}{3}$$, and so on.

(2) $$a_1-a_2=5$$. Substitute: $$a_1-\frac{a_{1}}{2}=5$$. We can solve for $$a_1$$ and thus will have the same case of knowing one term. Sufficient.

Can anybody reply how can we get the value of n here?

n is an index there. So, $$a_n$$ is nth term of the sequence.
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Intern  B
Joined: 19 Jun 2017
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Hi Bunuel,

Had the question instead been: "is the arithmetic mean of this sequence greater than 1/2?", would D still be the correct answer choice?

If NOT, kindly explain difference between the two questions. Also if possible, pls share any more examples which have M15-03's format (i.e. statements are sufficient because they provide the tools to solve if an additional variable -- in this case, last term in sequence -- was hypothetically provided).

Thank you
Math Expert V
Joined: 02 Sep 2009
Posts: 59573

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ulysses02 wrote:
Hi Bunuel,

Had the question instead been: "is the arithmetic mean of this sequence greater than 1/2?", would D still be the correct answer choice?

If NOT, kindly explain difference between the two questions. Also if possible, pls share any more examples which have M15-03's format (i.e. statements are sufficient because they provide the tools to solve if an additional variable -- in this case, last term in sequence -- was hypothetically provided).

Thank you

The mean = (the sum of the terms)/(the number of the terms). We don't know neither of those, so the answer will be E.

12. Sequences

Hope it helps.
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Intern  B
Joined: 19 Jun 2017
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Now I'm confused. Your initial solution said "...Notice that if we knew ANY term of the sequence we would be able to get all the other terms/numbers and thus answer the question."

Could you please explain how being able to determine all the other terms in the sequence from ANY term doesn't also enable one to determine the arithmetic mean, hypothetically speaking? Like what is the difference in sufficiency logic between both questions? Thanks

Bunuel wrote:
ulysses02 wrote:
Hi Bunuel,

Had the question instead been: "is the arithmetic mean of this sequence greater than 1/2?", would D still be the correct answer choice?

If NOT, kindly explain difference between the two questions. Also if possible, pls share any more examples which have M15-03's format (i.e. statements are sufficient because they provide the tools to solve if an additional variable -- in this case, last term in sequence -- was hypothetically provided).

Thank you

The mean = (the sum of the terms)/(the number of the terms). We don't know neither of those, so the answer will be E.

12. Sequences

Hope it helps.
Math Expert V
Joined: 02 Sep 2009
Posts: 59573

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ulysses02 wrote:
Now I'm confused. Your initial solution said "...Notice that if we knew ANY term of the sequence we would be able to get all the other terms/numbers and thus answer the question."

Could you please explain how being able to determine all the other terms in the sequence from ANY term doesn't also enable one to determine the arithmetic mean, hypothetically speaking? Like what is the difference in sufficiency logic between both questions? Thanks

Bunuel wrote:
ulysses02 wrote:
Hi Bunuel,

Had the question instead been: "is the arithmetic mean of this sequence greater than 1/2?", would D still be the correct answer choice?

If NOT, kindly explain difference between the two questions. Also if possible, pls share any more examples which have M15-03's format (i.e. statements are sufficient because they provide the tools to solve if an additional variable -- in this case, last term in sequence -- was hypothetically provided).

Thank you

The mean = (the sum of the terms)/(the number of the terms). We don't know neither of those, so the answer will be E.

12. Sequences

Hope it helps.

Yes, we can get any term of the sequence but we won't know how many terms in the sequence are there. For example, if we knew that there are 10 terms or 100 terms or 234 terms, then we could get the mean but if we don't know how many terms are there then how are we going to find the mean?
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Joined: 24 Oct 2015
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On finding the first 4-5 terms of the sequence -> 5, 5/3, 5/12, 1/12, 1/ 72... we find that the terms will keep becoming smaller and smaller as they get divided by 'n' -> hence we can say that there are 0 terms in the sequence greater than 1/2. Hence sufficient.
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Joined: 24 Jul 2019
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ShivaniD wrote:
On finding the first 4-5 terms of the sequence -> 5, 5/3, 5/12, 1/12, 1/ 72... we find that the terms will keep becoming smaller and smaller as they get divided by 'n' -> hence we can say that there are 0 terms in the sequence greater than 1/2. Hence sufficient.

I have the same missunderstanding with this question, Bunuel

Yes, with the data given we can figure out the terms in the sequence, but where does this question restrict the number of terms? The initital question is how many terms are > 1/2
My assumption was the following, even if we know A2 and therefore could calculate further values, what if the sequence just ends at a2? Then we would have two terms > 1/2
But what if the sequence ends at A3?

I dont question the validity of the question, I just don't read out of the statements the information that allows me to come up with one definite number for terms >1/2

Thanks
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# M15-03

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