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The value of 1/2 +(1/2)^2 + (1/2)^3.....+(1/2)^20 is between?

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The value of 1/2 +(1/2)^2 + (1/2)^3.....+(1/2)^20 is between?  [#permalink]

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New post 10 Jul 2011, 07:40
1
3
00:00
A
B
C
D
E

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  75% (hard)

Question Stats:

34% (01:44) correct 66% (01:27) wrong based on 88 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 \(\frac{3}{2}\)

M17-05

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Re: The value of 1/2 +(1/2)^2 + (1/2)^3.....+(1/2)^20 is between?  [#permalink]

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New post 10 Jul 2011, 07:44
This is a GP

So the sum is:

1/2(1 - (1/2)^19)/(1 - 1/2)

= 1 - (1/2)^19

Clearly this is > 9/10 (because (1/2)^19 is a very small number) and < 10/9 or 1

So D is the best choice
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Re: The value of 1/2 +(1/2)^2 + (1/2)^3.....+(1/2)^20 is between?  [#permalink]

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New post 10 Jul 2011, 07:47
AnkitK wrote:
What is 1/2 +(1/2)^2 + (1/2)^3.....+(1/2)^20 between?
A.1/2 and 2/3
B.2/3 and 3/4
C.3/4 and 9/10
D.9/10 and 10/9
E.10/9 and 3/2

What is shortest approach for this question?


the shortest way wud be t solve first 4/5 fraction
1/2 = 0.5
1/4 = 0.25
1/8 = 0.125
1/16 =.0625
1/32 = 0.03125
values after these becomes very small....

add these = 0.97
A,B ,C,E fails
only D is correct

PS: u can infact leave after 1/16... total = .93
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Re: The value of 1/2 +(1/2)^2 + (1/2)^3.....+(1/2)^20 is between?  [#permalink]

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New post 02 Nov 2011, 07:48
Pretty hard to solve under 2 mins. Should be D.
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Re: The value of 1/2 +(1/2)^2 + (1/2)^3.....+(1/2)^20 is between?  [#permalink]

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New post 02 Nov 2011, 08:27
Unless you know the sum formula for the geometric sequence, using sudhir18n's method would be easiest.
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Re: The value of 1/2 +(1/2)^2 + (1/2)^3.....+(1/2)^20 is between?  [#permalink]

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New post 23 Jan 2015, 09:58
[quote="AnkitK"]What is 1/2 +(1/2)^2 + (1/2)^3.....+(1/2)^20 between?
A.1/2 and 2/3
B.2/3 and 3/4
C.3/4 and 9/10
D.9/10 and 10/9
E.10/9 and 3/2


How can solve this guestion under 2 mins.
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Re: The value of 1/2 +(1/2)^2 + (1/2)^3.....+(1/2)^20 is between?  [#permalink]

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New post 23 Jan 2015, 09:59
AnkitK wrote:
What is 1/2 +(1/2)^2 + (1/2)^3.....+(1/2)^20 between?
A.1/2 and 2/3
B.2/3 and 3/4
C.3/4 and 9/10
D.9/10 and 10/9
E.10/9 and 3/2

What is shortest approach for this question?


How can solve this guestion under 2 mins
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Re: The value of 1/2 +(1/2)^2 + (1/2)^3.....+(1/2)^20 is between?  [#permalink]

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New post 23 Jan 2015, 11:21
AnkitK wrote:
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 \(\frac{3}{2}\)

M17-05


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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Re: The value of 1/2 +(1/2)^2 + (1/2)^3.....+(1/2)^20 is between?  [#permalink]

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New post 23 Jan 2015, 11:32
Tank you so much Bunuel
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Re: The value of 1/2 +(1/2)^2 + (1/2)^3.....+(1/2)^20 is between?  [#permalink]

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Re: The value of 1/2 +(1/2)^2 + (1/2)^3.....+(1/2)^20 is between?   [#permalink] 16 Jan 2019, 14:38
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