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Manager  Joined: 11 Feb 2011
Posts: 112
The value of 1/2 +(1/2)^2 + (1/2)^3.....+(1/2)^20 is between?  [#permalink]

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1
2 00:00

Difficulty:   75% (hard)

Question Stats: 28% (01:52) correct 72% (01:27) wrong based on 82 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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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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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
Manager  Joined: 28 Jul 2011
Posts: 89
Re: The value of 1/2 +(1/2)^2 + (1/2)^3.....+(1/2)^20 is between?  [#permalink]

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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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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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[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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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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Joined: 02 Sep 2009
Posts: 53798
Re: The value of 1/2 +(1/2)^2 + (1/2)^3.....+(1/2)^20 is between?  [#permalink]

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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.

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

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