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10 Jul 2011, 07:40
Kudos
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
59% (01:55) correct
41%
(01:48)
wrong
based on 400
sessions
History
Date
Time
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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
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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23 Jan 2015, 11:21
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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 \(\frac{3}{2}\)
The value of the given expression can be represented as the sum of a geometric sequence with the first term equal to \(\frac{1}{2}\) and the common ratio also equal to \(\frac{1}{2}\).
For an infinite geometric sequence with a common ratio \(|r| < 1\), the sum can be calculated as \(sum = \frac{b}{1-r}\), where \(b\) is the first term. So, if we had an infinite geometric sequence instead of just 20 terms, its sum would be \(Sum = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = 1\). This 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.
We can also use the direct formula for the sum of a finite geometric sequence:
For a geometric sequence 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 a very small number, \(1 - \frac{1}{2^{20}}\) will be less than 1 but very close to it.
Answer: D.
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 \(\frac{3}{2}\)
The value of the given expression can be represented as the sum of a geometric sequence with the first term equal to \(\frac{1}{2}\) and the common ratio also equal to \(\frac{1}{2}\).
For an infinite geometric sequence with a common ratio \(|r| < 1\), the sum can be calculated as \(sum = \frac{b}{1-r}\), where \(b\) is the first term. So, if we had an infinite geometric sequence instead of just 20 terms, its sum would be \(Sum = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = 1\). This 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.
We can also use the direct formula for the sum of a finite geometric sequence:
For a geometric sequence 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 a very small number, \(1 - \frac{1}{2^{20}}\) will be less than 1 but very close to it.
Answer: D.
General Discussion
10 Jul 2011, 07:44
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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
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
