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In a sequence a 1 , a 2 ,…, each term is defined as
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22 Jul 2019, 09:03
22 Jul 2019, 09:03
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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) \(1+\frac{1}{2^{11}}\)
(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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Re: In a sequence a 1 , a 2 ,…, each term is defined as
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23 Jul 2019, 06:42
23 Jul 2019, 06:42
4
Carcass 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) \(1+\frac{1}{2^{11}}\)
(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
Re: In a sequence a 1 , a 2 ,…, each term is defined as
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20 Aug 2019, 11:45
20 Aug 2019, 11:45
Or, as I'm sure you know, we could use the formula for the sum of a geometric sequence.
