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# M15-03

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Math Expert
Joined: 02 Sep 2009
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16 Sep 2014, 00:54
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Difficulty:

55% (hard)

Question Stats:

56% (01:27) correct 44% (01:49) wrong based on 84 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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16 Sep 2014, 00:54
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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11 Sep 2016, 05:41
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
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11 Sep 2016, 05:50
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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31 Dec 2016, 21:09
Wouldn't the caveat (for all n > 1) prevent you from solving for a1?
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01 Jan 2017, 08:17
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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01 Jan 2017, 10:52
So the formula is not true for n=1, right? In that case, how can you use it to determine what a1 is?
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07 Jan 2017, 23:04
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
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08 Jan 2017, 05:51
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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Joined: 06 Apr 2016
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17 Aug 2017, 17:45
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?
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17 Aug 2017, 21:21
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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20 Sep 2017, 18:00
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
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20 Sep 2017, 20:59
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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21 Sep 2017, 11:12
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
Joined: 02 Sep 2009
Posts: 47019

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21 Sep 2017, 11:19
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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Re: M15-03   [#permalink] 21 Sep 2017, 11:19
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# M15-03

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