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In the sequence above each term after the first one-half the previous term. If \(x\) is the tenth term of the sequence,

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\(\dfrac12, \dfrac14, \dfrac18, \dfrac1{16}, \dfrac1{32}, \ldots\)

In the sequence above each term after the first one-half the previous term. If \(x\) is the tenth term of the sequence, then \(x\) satisfies which of the following inequalities?

A) \(0.1 < x < 1\)
B) \(0.01 < x < 0.1\)
C) \(0.001 < x < 0.01\)
D) \(0.0001 < x < 0.001\)
E) \(0.00001 < x < 0.0001\)

[spoiler]OA=D[/spoiler]

Source: GMAT Prep

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VJesus12 wrote:
Tue Jun 30, 2020 8:08 am
\(\dfrac12, \dfrac14, \dfrac18, \dfrac1{16}, \dfrac1{32}, \ldots\)

In the sequence above each term after the first one-half the previous term. If \(x\) is the tenth term of the sequence, then \(x\) satisfies which of the following inequalities?

A) \(0.1 < x < 1\)
B) \(0.01 < x < 0.1\)
C) \(0.001 < x < 0.01\)
D) \(0.0001 < x < 0.001\)
E) \(0.00001 < x < 0.0001\)

[spoiler]OA=D[/spoiler]

Source: GMAT Prep
Given that the terms are \(\dfrac12, \dfrac14, \dfrac18, \dfrac1{16}, \dfrac1{32}, \ldots\), we can write them as

\(\dfrac12, \dfrac1{2^2}, \dfrac1{2^3}, \dfrac1{2^4}, \dfrac1{2^5}, \ldots\)

Thus, the 10th term = \( x = \dfrac1{2^{10}} = \dfrac1{1,024} < \left[\dfrac1{1,000} = \dfrac1{10^3} = 0.001 \right]\)

Similarly, the 10th term = \( x = \dfrac1{2^{10}} = \dfrac1{1,024} > \left[\dfrac1{10,000} = \dfrac1{10^4} = 0.0001 \right]\)

The correct answer is D.

Correct answer: D

Hope this helps!

-Jay
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