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

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Joined: 02 Sep 2009
Posts: 58954
In a sequence a1, a2, …, each term is defined as an = 1/2^n. Which of  [#permalink]

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25 Jul 2019, 23:44
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65% (hard)

Question Stats:

48% (01:49) correct 53% (02:17) wrong based on 40 sessions

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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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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.

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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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20 Oct 2019, 07:05
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Top Contributor
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).

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 $$1-\frac{1}{2^{10}}$$

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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