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Math Expert V
Joined: 02 Sep 2009
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Difficulty:   65% (hard)

Question Stats: 56% (01:59) correct 44% (01:33) wrong based on 79 sessions

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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 $$\frac{3}{2}$$

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Math Expert V
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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.

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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

Originally posted by Raihanuddin on 27 Nov 2014, 16:19.
Last edited by Raihanuddin on 09 Nov 2015, 09:28, edited 1 time in total.
Verbal Forum Moderator Joined: 15 Apr 2013
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Dear Buenel,

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

Math Expert V
Joined: 02 Sep 2009
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WillGetIt wrote:
Dear Buenel,

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

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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Thanks for quick feedback.

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

Math Expert V
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WillGetIt wrote:
Thanks for which feedback.

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

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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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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is GP even asked in GMAT?
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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. Intern  B
Joined: 22 Nov 2014
Posts: 29

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shashanksagar wrote:
is GP even asked in GMAT?

havent seen a single sum
Intern  B
Joined: 26 May 2016
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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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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.
Math Expert V
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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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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?

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
Math Expert V
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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?

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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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?

Intern  B
Joined: 08 Sep 2016
Posts: 32

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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. 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?

Intern  B
Joined: 17 Nov 2016
Posts: 2

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I did it by approximation-
20*(1/2)^10= 5/256
=1/11
=.09
hence option D Re: M17-05   [#permalink] 27 Nov 2017, 09:18

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