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The limiting sum of the infinite series [m][fraction]1/2[/fraction] +

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The limiting sum of the infinite series [m][fraction]1/2[/fraction] +  [#permalink]

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New post Updated on: 22 May 2020, 13:12
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The limiting sum of the infinite series \(\frac{1}{2} + \frac{3}{4} + \frac{5}{8} + \frac{7}{16} + \frac{9}{32}\)+ ... whose \(n\)th term is \(\frac{2n-1}{2^n}\) is:
A) \(\frac{5}{2}\)
B) 3
C) \(\frac{7}{2}\)
D) \(\frac{15}{4}\)
E) 4

Source: Quantum GMAT

Originally posted by TheUltimateWinner on 22 May 2020, 08:24.
Last edited by TheUltimateWinner on 22 May 2020, 13:12, edited 1 time in total.
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Re: The limiting sum of the infinite series [m][fraction]1/2[/fraction] +  [#permalink]

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New post 22 May 2020, 12:58
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Asad wrote:
The limiting sum of the infinite series \(\frac{1}{2} + \frac{3}{4} + \frac{5}{8} + \frac{7}{16} + \frac{9}{31}\)+ ... whose \(n\)th term is \(\frac{2n-1}{2^n}\) is:
A) \(\frac{5}{2}\)
B) 3
C) \(\frac{7}{2}\)
D) \(\frac{15}{4}\)
E) 4

Source: Quantum GMAT

Hello, Asad. Thank you for posting this question, but I think the fifth term should be \(\frac{9}{32}\) instead. It is not that I think anyone solving the question would be flummoxed by the change, but for the sake of consistency with the established rule, I thought I would bring the issue to your attention.

- Andrew
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The limiting sum of the infinite series [m][fraction]1/2[/fraction] +  [#permalink]

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New post 22 May 2020, 13:15
MentorTutoring wrote:
Asad wrote:
The limiting sum of the infinite series \(\frac{1}{2} + \frac{3}{4} + \frac{5}{8} + \frac{7}{16} + \frac{9}{31}\)+ ... whose \(n\)th term is \(\frac{2n-1}{2^n}\) is:
A) \(\frac{5}{2}\)
B) 3
C) \(\frac{7}{2}\)
D) \(\frac{15}{4}\)
E) 4

Source: Quantum GMAT

Hello, Asad. Thank you for posting this question, but I think the fifth term should be \(\frac{9}{32}\) instead. It is not that I think anyone solving the question would be flummoxed by the change, but for the sake of consistency with the established rule, I thought I would bring the issue to your attention.

- Andrew

MentorTutoring
You're right. Edited.....
Thank you...
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The limiting sum of the infinite series [m][fraction]1/2[/fraction] +  [#permalink]

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New post 28 May 2020, 12:40
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Hello again, Asad. I wanted to outline my approach to the question, in case it differs from your own or that of any onlooker. I typically favor a more intuitive approach than a rigorous mathematical approach. In the question at hand, I started by converting all the fractions to thirty-seconds and adding:

\(\frac{16}{32}+\frac{24}{32}+\frac{20}{32}+\frac{14}{32}+\frac{9}{32}=\frac{83}{32}\)

Now, our next fraction would be in sixty-fourths, per the pattern, and the numerator would be 11. To put this value together with the earlier sum in a meaningful way, I converted the earlier answer to sixty-fourths:

\(\frac{11}{64}+\frac{166}{64}=\frac{177}{64}\)

At this point, we have a decimal value that would lie between 2 and 3, but closer to the 3 end, since

\(64*3=192\)

Looking at the answer choices, we can eliminate (A) since that is already too low. How do we know?

\(64*2+32<177\)

We can also appreciate, without doing more math, that the sum will grow in smaller and smaller increments, and our last term was already about one-sixth. Thus, it appears as though our sum will converge on 3. Still, for the inner fact-checker, we could go on. In terms of sixty-fourths, we have 15 more to add in the numerator (to get from 177 to 192), and that is going to take multiple steps. The next two terms would give us

\(\frac{13}{128}+\frac{15}{256}\)

or, in terms of sixty-fourths, the numerators would be

\(\frac{13}{2}+\frac{15}{4}=\frac{41}{4}\)

Closer still, as we would have just 4.75 more to go to get our numerator to 192. But to be honest, I see no way that, in keeping with this pattern, the numerator will increase to 192 and another 32 (half of 64) to get us up to answer (C). Although I would have selected my answer already, if I had not before, I would feel confident choosing (B), confirming, and moving on to the next challenge.

Thank you for sharing the question. What is the source, if you do not mind my asking?

- Andrew
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The limiting sum of the infinite series [m][fraction]1/2[/fraction] +  [#permalink]

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New post 28 May 2020, 12:55
The source is 'Quantum GMAT'... Already source mentioned in the spoiler. This question has been designed for them who are targeting Q51! Thanks..
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Re: The limiting sum of the infinite series [m][fraction]1/2[/fraction] +  [#permalink]

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New post 28 May 2020, 14:27
2
Asad wrote:
The source is 'Quantum GMAT'... Already source mentioned in the spoiler. This question has been designed for them who are targeting Q51! Thanks..

Ah, the spoiler that I sometimes cannot get to work. I am not sure if it is a browser issue or a computer issue, but sometimes I click on the spoiler or the OA/OE and get nothing. It worked this time, so I feel foolish. (Oh well.) I would like to hit a 51 in Quant myself. Maybe later this year, once matters settle down, I will put myself to the test and get a Verified Score to post... hopefully with a pair of 51s to show for the effort. Time will tell.

- Andrew
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The limiting sum of the infinite series [m][fraction]1/2[/fraction] +  [#permalink]

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New post 28 May 2020, 14:46
1
MentorTutoring wrote:
Asad wrote:
The source is 'Quantum GMAT'... Already source mentioned in the spoiler. This question has been designed for them who are targeting Q51! Thanks..

Ah, the spoiler that I sometimes cannot get to work. I am not sure if it is a browser issue or a computer issue, but sometimes I click on the spoiler or the OA/OE and get nothing. It worked this time, so I feel foolish. (Oh well.) I would like to hit a 51 in Quant myself. Maybe later this year, once matters settle down, I will put myself to the test and get a Verified Score to post... hopefully with a pair of 51s to show for the effort. Time will tell.

- Andrew

good luck for you so that you can get q51! in the mean time you can see the explanation from the following link (if you need..)

https://www.facebook.com/gmatquantum/vi ... 560222132/
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Re: The limiting sum of the infinite series [m][fraction]1/2[/fraction] +  [#permalink]

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New post 28 May 2020, 15:09
1
Asad wrote:
MentorTutoring wrote:
Asad wrote:
The source is 'Quantum GMAT'... Already source mentioned in the spoiler. This question has been designed for them who are targeting Q51! Thanks..

Ah, the spoiler that I sometimes cannot get to work. I am not sure if it is a browser issue or a computer issue, but sometimes I click on the spoiler or the OA/OE and get nothing. It worked this time, so I feel foolish. (Oh well.) I would like to hit a 51 in Quant myself. Maybe later this year, once matters settle down, I will put myself to the test and get a Verified Score to post... hopefully with a pair of 51s to show for the effort. Time will tell.

- Andrew

good luck for you so that you can get q51! in the mean time you can see the explanation from the following link (if you need..)

https://www.facebook.com/gmatquantum/vi ... 560222132/

Thank you for the well wishes, as well as for the link. (I am not on Facebook, so I would not have found anything on my own that way.) Kudos from me for all the help.

- Andrew
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Re: The limiting sum of the infinite series [m][fraction]1/2[/fraction] +  [#permalink]

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New post 28 May 2020, 15:18
MentorTutoring wrote:
Asad wrote:
MentorTutoring wrote:
Asad wrote:
The source is 'Quantum GMAT'... Already source mentioned in the spoiler. This question has been designed for them who are targeting Q51! Thanks..

Ah, the spoiler that I sometimes cannot get to work. I am not sure if it is a browser issue or a computer issue, but sometimes I click on the spoiler or the OA/OE and get nothing. It worked this time, so I feel foolish. (Oh well.) I would like to hit a 51 in Quant myself. Maybe later this year, once matters settle down, I will put myself to the test and get a Verified Score to post... hopefully with a pair of 51s to show for the effort. Time will tell.

- Andrew

good luck for you so that you can get q51! in the mean time you can see the explanation from the following link (if you need..)

https://www.facebook.com/gmatquantum/vi ... 560222132/

Thank you for the well wishes, as well as for the link. (I am not on Facebook, so I would not have found anything on my own that way.) Kudos from me for all the help.

- Andrew

I think you should not have any facebook ID to watch that video! if you click on this link it'll be automatically started..
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Re: The limiting sum of the infinite series [m][fraction]1/2[/fraction] +   [#permalink] 28 May 2020, 15:18

The limiting sum of the infinite series [m][fraction]1/2[/fraction] +

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