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In a sequence a1, a2, …, each term is defined as an = 1/2^n. Which of

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In a sequence a1, a2, …, each term is defined as an = 1/2^n. Which of  [#permalink]

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New post 25 Jul 2019, 23:44
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A
B
C
D
E

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Question Stats:

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In a sequence \(a_1\), \(a_2\), …, each term is defined as \(a_n=\frac{1}{2^n}\). Which of the following expressions represents the sum of the first 10 terms of \(a_n\) ?

(A) \(1-\frac{1}{2^{10}}\)

(B) \(1- \frac{1}{2^9}\)

(C) \(1+\frac{1}{2^9}\)

(D) \(1+\frac{1}{2^{10}}\)

(E) \(1+\frac{1}{2^{11}}\)

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In a sequence a1, a2, …, each term is defined as an = 1/2^n. Which of  [#permalink]

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New post 26 Jul 2019, 01:19

Solution


Given:
    • In a sequence, \(a_n = \frac{1}{2^n}\)

To find:
    • The sum of first 10 terms

Approach and Working Out:
    • a = \(\frac{1}{2}\), n = 10, and r = \(\frac{1}{2}\)

\(S_{10} = \frac{a * ( 1- r^n)}{1 - r}\)
    • Sum of first ten terms = \(\frac{1}{2} + \frac{1}{2^2} + …. + \frac{1}{2^{10}} = \frac{1}{2} * (1 – \frac{1}{2^{10}})/(1 – \frac{1}{2}) = 1 – \frac{1}{2^{10}}\)

Hence, the correct answer is Option A.

Answer: A

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Re: In a sequence a1, a2, …, each term is defined as an = 1/2^n. Which of  [#permalink]

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New post 20 Oct 2019, 07:05
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Bunuel wrote:
In a sequence \(a_1\), \(a_2\), …, each term is defined as \(a_n=\frac{1}{2^n}\). Which of the following expressions represents the sum of the first 10 terms of \(a_n\) ?

(A) \(1-\frac{1}{2^{10}}\)

(B) \(1- \frac{1}{2^9}\)

(C) \(1+\frac{1}{2^9}\)

(D) \(1+\frac{1}{2^{10}}\)

(E) \(1+\frac{1}{2^{11}}\)


Let's look for a pattern

If \(n=1\), the sequence is \(\frac{1}{2}\). So the SUM = \(\frac{1}{2}\)

If \(n=2\), the sequence is \(\frac{1}{2}\), \(\frac{1}{4}\). So the SUM = \(\frac{3}{4}\)

If \(n=3\), the sequence is \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{1}{8}\). So the SUM = \(\frac{7}{8}\)

If \(n=4\), the sequence is \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{1}{8}\), \(\frac{1}{16}\). So the SUM = \(\frac{15}{16}\)

At this point, we may notice that the SUM is always less than 1.
So, we can ELIMINATE C, D and E

Now examine the remaining answer choices (A and B).
This gives us a hint about the correct answer.

Notice that, when we have \(n\) terms, the SUM is always \(\frac{1}{2^n}\) LESS THAN 1

The last term in the given sequence is \(\frac{1}{2^{10}}\)

So, the sum will equal \(1-\frac{1}{2^{10}}\)

Answer: A

Cheers,
Brent
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Re: In a sequence a1, a2, …, each term is defined as an = 1/2^n. Which of   [#permalink] 20 Oct 2019, 07:05
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