we got k = 1 to 10

our number looks like (-1)^K+1 x (1/2)^K

for k = 1 we get n1 = (-1)^2 x(1/2)^1 = (1)x(1/2) = 1/2
for k = 2 we get n2 = (-1)^3 x(1/2)^2 = (-1)x(1/4) = -1/4

so adding these two we get result as 1/4 .

we can make pairs like n1 and n2 , n3 and n4 and so on.

so the sums will be ...

1/4 + 1/16 + 1/64 + 1/256 + 1/1024

which cant cross 1/2

so its between 1/4 and 1/2

this is probably not the best way to explain stuff but hope it helps.
_________________

The Legion dies, it does not surrender.

Your answer is correct but remember that GMAT is testing us also in speed. So we must proceed as fast as possible. This kind of questions when doing all calculation take too much time and knowing some basic rules helps eliminate such waste.

OK, let's deal with this problem with some knowledge of basic math and common sense.

First of all we see that there is set of 10 numbers and every even term is negative.

Second it's not hard to get this numbers: 1/2, -1/4, 1/8, -1/16/, 1/32... enough for calculations, we see pattern now.

And now the main part: adding them up is quite a job, after calculations you'll get 341/1024. You can add them up by pairs but it's also time consuming. Once we've done it we can conclude that it's more than 1/4 and less than 1/2, so answer is D

BUT there is another way:

We have two pairs:
1/2, 1/8... (5 terms)
-1/4, -1/16 (5 terms)
These pairs are decreasing by 1/4 each time.

THE RULE: when we have such sequence which is decreasing by constant<1 the sum of theese sequence tends to become first term/(1-constant) (in our case constant is 1/4). (tends means that if we have infinite number of terms then the sum would be...)

For first pair of numbers we'll have 1/2/(1-1/4)=2/3
For second pair of numbers we'll have -1/4(1-1/4)=-1/3
Add them up 2/3-1/3=1/3.

This means that no matter how many number we have their sum never will be more then 1/3 (A.B and C are out). Also means that the sum of our sequence is very close to 1/3 and for sure more than 1/4 (E out). So answer is D
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