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The value of \(\frac{1}{2} + (\frac{1}{2})^2 + (\frac{1}{2})^3 + ... + (\frac{1}{2})^{20}\) is between?

A. \(\frac{1}{2}\) and \(\frac{2}{3}\)
B. \(\frac{2}{3}\) and \(\frac{3}{4}\)
C. \(\frac{3}{4}\) and \(\frac{9}{10}\)
D. \(\frac{9}{10}\) and \(\frac{10}{9}\)
E. \(\frac{10}{9}\) and \(\frac{3}{2}\)
[Reveal] Spoiler: OA

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Official Solution:

The value of \(\frac{1}{2} + (\frac{1}{2})^2 + (\frac{1}{2})^3 + ... + (\frac{1}{2})^{20}\) is between?

A. \(\frac{1}{2}\) and \(\frac{2}{3}\)
B. \(\frac{2}{3}\) and \(\frac{3}{4}\)
C. \(\frac{3}{4}\) and \(\frac{9}{10}\)
D. \(\frac{9}{10}\) and \(\frac{10}{9}\)
E. \(\frac{10}{9}\) and \(\frac{3}{2}\)


We have the sum of a geometric progression with the first term equal to \(\frac{1}{2}\) and the common ratio also equal to \(\frac{1}{2}\).

Now, the sum of infinite geometric progression with common ratio \(|r| \lt 1\), is \(sum=\frac{b}{1-r}\), where \(b\) is the first term. So, if we had infinite geometric progression instead of just 20 terms then its sum would be \(Sum=\frac{\frac{1}{2}}{1-\frac{1}{2}}=1\). Which means that the sum of this sequence will never exceed 1. Also since we have a large enough number of terms (20), the sum will be very close to 1, so we can safely choose answer choice D.

One can also use direct formula.

We have geometric progression with \(b=\frac{1}{2}\), \(r=\frac{1}{2}\) and \(n=20\);

\(S_n=\frac{b(1-r^n)}{(1-r)}\), so:

\(S_{20}=\frac{\frac{1}{2}(1-\frac{1}{2^{20}})}{(1-\frac{1}{2})}=1-\frac{1}{2^{20}}\). Since \(\frac{1}{2^{20}}\) is very small number then \(1-\frac{1}{2^{20}}\) will be less than 1 but very close to it.


Answer: D
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In case we forget formula we can do it by following approximation
1/2=.50
1/2^2 =.250
1/2^3= .125
.............
.50+.250+.125 = .875
So we are sure that by other decimal number value we will be very close to 1 but less than 1

Only option D shows that

Last edited by Raihanuddin on 09 Nov 2015, 08:28, edited 1 time in total.

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New post 21 Aug 2015, 19:38
Dear Buenel,

As per your explanation, answer should be C not D.

In (D) 10/9 is actually greater than 01.

Please suggest if I am wrong. Please assist.

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New post 22 Aug 2015, 03:58
WillGetIt wrote:
Dear Buenel,

As per your explanation, answer should be C not D.

In (D) 10/9 is actually greater than 01.

Please suggest if I am wrong. Please assist.


Yes, you are wrong.

The value of the expression is between 9/10 and 10/9 (D). It's very slightly less than 1.
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New post 22 Aug 2015, 04:02
Thanks for quick feedback.

The whole issue with D was that 10/9 is 1.11... Which is greater than 1?

Please clarify

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New post 22 Aug 2015, 04:05
WillGetIt wrote:
Thanks for which feedback.

The whole issue with D was that 10/9 is 1.11... Which is greater than 1?

Please clarify


Note that the question asks about the range, not the exact value of the expression. If the value of the expression were 0.9999999 wouldn't it still be correct to say that it's between 0.9 and 1000000000000000000000000?
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New post 22 Aug 2015, 04:17
OK

Thanks a lot

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I actually forgot the formula when I had this question so I did the most native manual way I can:
1/2 + 1/4 = 3/4 + 1/8 = 7/8 + 1/16 = 15/16
Stop here: 15/16 > 9/10 and since the rhythm of the series we can see that the sum will always <1
--> correct ans: D
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New post 09 Jul 2016, 13:46
is GP even asked in GMAT?

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New post 08 Oct 2016, 15:26
Linhbiz wrote:
I actually forgot the formula when I had this question so I did the most native manual way I can:
1/2 + 1/4 = 3/4 + 1/8 = 7/8 + 1/16 = 15/16
Stop here: 15/16 > 9/10 and since the rhythm of the series we can see that the sum will always <1
--> correct ans: D


I didn't even know the formula but took the same approach. I was able to solve the problem within 1 minute. :-D

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New post 22 Dec 2016, 05:16
shashanksagar wrote:
is GP even asked in GMAT?




havent seen a single sum

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New post 04 May 2017, 03:04
Raihanuddin wrote:
In case we forget formula we can do it by following approximation
1/2=.50
1/2^2 =.250
1/2^3= .125
.............
.50+.250+.125 = .875
So we are sure that by other decimal number value we will be very close to 1 but less than 1

Only option D shows that


To be sure, I added 1/2, 1/4,1/8 and 1/16 to to get 0.875 + 0.0625 > 0.93

and since the further we go, the smaller fractions we add, we should be close to 0.99
Hence D

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New post 16 Jun 2017, 05:28
I agree that option D is the closest in terms of the correct answer,but the range is wrong as 10/9 is greater than 1. So I had eliminated that option at first glance.
I think none of the options in this question is correct.

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New post 16 Jun 2017, 05:33
Pitabdhi wrote:
I agree that option D is the closest in terms of the correct answer,but the range is wrong as 10/9 is greater than 1. So I had eliminated that option at first glance.
I think none of the options in this question is correct.


Please re-read the solution and the discussion above. The value of the expression is between 9/10 and 10/9 (D). It's very slightly less than 1.
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New post 28 Jun 2017, 18:21
Bunuel wrote:
WillGetIt wrote:
Thanks for which feedback.

The whole issue with D was that 10/9 is 1.11... Which is greater than 1?

Please clarify


Note that the question asks about the range, not the exact value of the expression. If the value of the expression were 0.9999999 wouldn't it still be correct to say that it's between 0.9 and 1000000000000000000000000?


Bunuel, yes the values can be considered that way. But also we in this range there are values which are greater than 1 and we know for sure it cannot be greater than 1. How do we justify that...what if the value of the expression is 1.00000000009 or 1.10, wouldn't it be greater than 1? Pls clarify

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New post 28 Jun 2017, 20:21
Chets25 wrote:
Bunuel wrote:
WillGetIt wrote:
Thanks for which feedback.

The whole issue with D was that 10/9 is 1.11... Which is greater than 1?

Please clarify


Note that the question asks about the range, not the exact value of the expression. If the value of the expression were 0.9999999 wouldn't it still be correct to say that it's between 0.9 and 1000000000000000000000000?


Bunuel, yes the values can be considered that way. But also we in this range there are values which are greater than 1 and we know for sure it cannot be greater than 1. How do we justify that...what if the value of the expression is 1.00000000009 or 1.10, wouldn't it be greater than 1? Pls clarify


How does this matter?

Say x = 10. Wouldn't it be correct to say that it's between -1000000 and 10000000000000?
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New post 30 Jun 2017, 03:46
Hi,

I understood that you did it without using the formula...but can u please explain how did u arrive ta the answer in details

I understood until point u get 15/16..after that how did u arrive at the answer?

Thanks in advance!

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New post 30 Jun 2017, 03:47
Rookie84 wrote:
Linhbiz wrote:
I actually forgot the formula when I had this question so I did the most native manual way I can:
1/2 + 1/4 = 3/4 + 1/8 = 7/8 + 1/16 = 15/16
Stop here: 15/16 > 9/10 and since the rhythm of the series we can see that the sum will always <1
--> correct ans: D


I didn't even know the formula but took the same approach. I was able to solve the problem within 1 minute. :-D


Hi,

I understood that you did it without using the formula...but can u please explain how did u arrive ta the answer in details

I understood until point u get 15/16..after that how did u arrive at the answer?

Thanks in advance!

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New post 27 Nov 2017, 08:18
I did it by approximation-
20*(1/2)^10= 5/256
=1/11
=.09
hence option D

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Re: M17-05   [#permalink] 27 Nov 2017, 08:18
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