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sum of terms problem - please help explain

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by kjallow » Tue Feb 28, 2012 7:16 pm
Question: For every integer K from 1-10, inclusive, the Kth term of a certain sequence is given by (-1)^k+1(1/2^k). If T is the sum of the first 10 terms in the sequence, then T is:
a) greater than 2
b) between 1 and 2
c) between 1/2 and 1
d) between 1/4 and 1/2
e) less than 1/4

I tried using the formula below but just don't get it
Sum = average * # of terms
where average = (1st term + last term)/2

official answer is d - between 1/4 and 1/2
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by GMATGuruNY » Tue Feb 28, 2012 7:24 pm
I posted a solution here:

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by krusta80 » Tue Feb 28, 2012 8:15 pm
kjallow wrote:Question: For every integer K from 1-10, inclusive, the Kth term of a certain sequence is given by (-1)^k+1(1/2^k). If T is the sum of the first 10 terms in the sequence, then T is:
a) greater than 2
b) between 1 and 2
c) between 1/2 and 1
d) between 1/4 and 1/2
e) less than 1/4

I tried using the formula below but just don't get it
Sum = average * # of terms
where average = (1st term + last term)/2

official answer is d - between 1/4 and 1/2
Looks like you copied it wrong?
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by Anurag@Gurome » Tue Feb 28, 2012 9:32 pm
kjallow wrote:Question: For every integer K from 1-10, inclusive, the Kth term of a certain sequence is given by (-1)^k+1(1/2^k). If T is the sum of the first 10 terms in the sequence, then T is:
a) greater than 2
b) between 1 and 2
c) between 1/2 and 1
d) between 1/4 and 1/2
e) less than 1/4

I tried using the formula below but just don't get it
Sum = average * # of terms
where average = (1st term + last term)/2

official answer is d - between 1/4 and 1/2
We know that the sum of n terms of a geometric series is given by:
S(n) = a(1 - r�)(1 - r), where a is the first term, r is the common ratio of the geometric progression and n = number of terms.

Here, a = 1/2, r = -1/2, n = 10
T = 1/2[1 - (-1/2)^10]/[1 + 1/2]
= 1/2[1 - 1/1024]/[3/2]
= 1/2 * 1023/1024 * (2/3)
= (1023/1024) * (1/3)
Now (1023/1024) = 1 approx, so T = 1/3, which lies between 1/4 and 1/2.

The correct answer is D.
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by kjallow » Thu Mar 01, 2012 1:30 pm
Anurag@Gurome wrote:
kjallow wrote:Question: For every integer K from 1-10, inclusive, the Kth term of a certain sequence is given by (-1)^k+1(1/2^k). If T is the sum of the first 10 terms in the sequence, then T is:
a) greater than 2
b) between 1 and 2
c) between 1/2 and 1
d) between 1/4 and 1/2
e) less than 1/4

I tried using the formula below but just don't get it
Sum = average * # of terms
where average = (1st term + last term)/2

official answer is d - between 1/4 and 1/2
We know that the sum of n terms of a geometric series is given by:
S(n) = a(1 - r�)(1 - r), where a is the first term, r is the common ratio of the geometric progression and n = number of terms.

Here, a = 1/2, r = -1/2, n = 10
T = 1/2[1 - (-1/2)^10]/[1 + 1/2]
= 1/2[1 - 1/1024]/[3/2]
= 1/2 * 1023/1024 * (2/3)
= (1023/1024) * (1/3)
Now (1023/1024) = 1 approx, so T = 1/3, which lies between 1/4 and 1/2.

The correct answer is D.
Thanks Anurag, I like the formula approach. Can you please explain though why r is -1/2 and not just 1/2? From this am assuming too that it helps to know before test day that 2^10 is 1024 instead of having to compute it then. Thanks again!
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