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Sequence Question - Please help! Thanks in advance!

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Sequence Question - Please help! Thanks in advance!

by Tdot23 » Wed May 23, 2012 4:06 am
Hi BTG Community,
Could someone please take a look at the following question I've came across in the GMAC test:

Q: For every integer k from 1 to 10, kth term is (-1)^k+1 (1/2^k). If T equals the sum of first 10 terms, T is?

A: Between 1/4 and 1/2

I got to the answer (eventually) but I'm not confident in the way I got there so if you could kindly post a clear way to solving this, it would be much appreciated. Thank.
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by GMATGuruNY » Wed May 23, 2012 4:27 am
NYC23 wrote:Hi BTG Community,
Could someone please take a look at the following question I've came across in the GMAC test:

Q: For every integer k from 1 to 10, kth term is (-1)^k+1 (1/2^k). If T equals the sum of first 10 terms, T is?

A: Between 1/4 and 1/2

I got to the answer (eventually) but I'm not confident in the way I got there so if you could kindly post a clear way to solving this, it would be much appreciated. Thank.
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by Anurag@Gurome » Wed May 23, 2012 4:31 am
NYC23 wrote:Hi BTG Community,
Could someone please take a look at the following question I've came across in the GMAC test:

Q: For every integer k from 1 to 10, kth term is (-1)^k+1 (1/2^k). If T equals the sum of first 10 terms, T is?

A: Between 1/4 and 1/2

I got to the answer (eventually) but I'm not confident in the way I got there so if you could kindly post a clear way to solving this, it would be much appreciated. Thank.
Question is: For every integer k from 1 to 10, kth term of a certain sequence can be written as [(-1)^(k+1)]*(1/2^k). If T is the sum of 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

1st term : [(-1)^(1 + 1)]*(1/2^1) = 1/2
2nd term : [(-1)^(2 + 1)]*(1/2^2) = -1/4
3rd term : [(-1)^(3 + 1)]*(1/2^3) = 1/8

Hence, the terms are 1/2, -1/4, 1/8, -1/16, ... etc.

Now, we see that the first term is 1/2 and after that we alternately subtract and add half of the previous term. Hence, there is no way the sum of the terms will ever be greater than 1/2. So first three options are discarded.

Now, note that sum of first two terms is 1/4 and after that we alternately add and subtract half of the previous term. Hence we make the sum greater than 1/4 by 1/8 and then decrease it by 1/16 and so on. Hence the sum is never going to be less than 1/4. So last option is also discarded.

The correct answer is D.
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