Rosh,
How I did it......
k=1 -> 1/2
k=2 -> -1/4
k=3 -> 1/8
k=4 -> -1/16
....
....
So the sum will be 1/2-1/4+1/8-1/16+1/32-1/64+1/128-1/256+1/512-1/1024
The same as 1/4+1/16+1/64+1/128+1/256+1/1024=(256+64+16+8+4+1)/1024=349/1024, which is less than 1/2 but greater than 1/4.
Answer must be D.
BTW: To beat the GMAT you have to know all the powers of two from 2^2 to 2^10. I believe the experts will show you a much quicker way to find the solution algebrically.
Sequence
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Ian Stewart
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The question is closely related to one of the famous problems in ancient mathematics- one of Zeno's paradoxes. If you look at this sum:
1/2 + 1/2^2 + 1/2^3 + 1/2^4 + ...
which is the same as:
1/2 + 1/4 + 1/8 + 1/16 + ...
this sum gets closer and closer to one the more terms you add, but never gets greater than one. There are algebraic ways to see this, but Zeno described running a race. First you must cover 1/2 the distance. Then you must cover 1/2 of the remaining distance, or 1/4 of the total distance. Then you must cover 1/2 of the remaining distance, or 1/8 of the total. And so on. So these fractions must add to something less than (but very close to) 1, unless you keep adding forever. (That's something you learn how to do in calculus, by the way
)
So when I saw this question, as with every sequence question, I wrote down a few terms:
1/2 - 1/4 + 1/8 - 1/16 +...
then noticed this will become
1/4 + 1/16 + 1/64 +...
then noticed that this will be greater than 1/4, but also noticed that this must be less than
1/4 + 1/8 + 1/16 + 1/32 +...
because we're leaving a few things out. But this sum I've just written is less than 1/2, by the same logic as used in the Zeno example above.
It's worth looking at the answer choices here- you know you don't need an exact value, so there will be a faster way to arrive at an answer here. If I hadn't noticed the above, I would still have done a quick estimate. In the sum 1/4 + 1/16 + 1/64 + 1/256 + 1/1024 the last few terms are minuscule. I'd only bother looking at them if the first few terms added to something very close to 1/2, and they don't, so there is only one possible answer- D.
1/2 + 1/2^2 + 1/2^3 + 1/2^4 + ...
which is the same as:
1/2 + 1/4 + 1/8 + 1/16 + ...
this sum gets closer and closer to one the more terms you add, but never gets greater than one. There are algebraic ways to see this, but Zeno described running a race. First you must cover 1/2 the distance. Then you must cover 1/2 of the remaining distance, or 1/4 of the total distance. Then you must cover 1/2 of the remaining distance, or 1/8 of the total. And so on. So these fractions must add to something less than (but very close to) 1, unless you keep adding forever. (That's something you learn how to do in calculus, by the way
)So when I saw this question, as with every sequence question, I wrote down a few terms:
1/2 - 1/4 + 1/8 - 1/16 +...
then noticed this will become
1/4 + 1/16 + 1/64 +...
then noticed that this will be greater than 1/4, but also noticed that this must be less than
1/4 + 1/8 + 1/16 + 1/32 +...
because we're leaving a few things out. But this sum I've just written is less than 1/2, by the same logic as used in the Zeno example above.
It's worth looking at the answer choices here- you know you don't need an exact value, so there will be a faster way to arrive at an answer here. If I hadn't noticed the above, I would still have done a quick estimate. In the sum 1/4 + 1/16 + 1/64 + 1/256 + 1/1024 the last few terms are minuscule. I'd only bother looking at them if the first few terms added to something very close to 1/2, and they don't, so there is only one possible answer- D.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com
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