For every integer k from 1 to 10, inclusive the "k"th term of a certain sequence is given by (-1)^{(k+1)}*(\frac{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 don't know how to solve it. it goes as (1/2)-(1/2^2)+(1/2^3)- ... -(1/2^10)=T

A few things here:

* I've never seen a real GMAT question that requires one to know any geometric sequence formulas. Of course there are questions where you might use such formulas, but there will always be a different approach available;

* In any sequence question which gives an expression for each term, you'll always want to write down the first few terms using the given expression to work out what the sequence looks like. Here we have:

\frac{1}{2}, \frac{-1}{4}, \frac{1}{8}, \frac{-1}{16}, \frac{1}{32}, \frac{-1}{64}, \frac{1}{128}, \frac{-1}{256}, \frac{1}{512}, \frac{-1}{1024}

* Now, adding all of these fractions together would take a long time to do. The GMAT *never* requires you to perform any crazy calculations, so there must be a different way to answer the question. Notice the answer choices are only estimates, so we only need to estimate the sum of the first 10 terms. When we want to estimate the value of a sum, we ignore terms that make only a tiny contribution to the sum. The last few terms in our sequence are minuscule compared to the first few, so to get a good estimate, we can completely ignore them; adding, say, the first four terms will give a perfectly good approximation of the sum here (you get 5/16, which is enough to choose the right answer).

* The sequence in this question is what is known as an 'alternating sequence' -- that is, the terms alternate between positive and negative values. When adding an alternating sequence, you most often want to add your terms in pairs first, grouping one positive and one negative (add the 1st and 2nd term, the 3rd and 4th, and so on). One doesn't need to do this for this question, but it does make the answer a bit easier to see - we'd find our sum is

\frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \frac{1}{256} + \frac{1}{1024}

from which you can instantly see the sum is greater than 1/4. Since every term here is tiny after the first two, the sum is certainly less than 1/2.
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the way i saw it is this way, the first two terms are 1/2 and -1/4, added together will give you 1/4. the rest of the term pairs will only be added to this, so there's no way that the end result will be less than 1/4. you now have your lower-bound...

-----> if you're pressed for time, you'll have a pretty good idea of the answer because D is the only one that has greater than 1/4, you can intelligently guess that... if you do have more time, press on...


if you look at the other term pairs, they'll be: 1/8, 1/16, 1/32 ... even if this goes on forever, the sequence will keep on halving itself...

now you do some quick calculations:

1/4 =2/8 = 4/16 = 8/32 .... etc.
+ 1/8 = 2/16 = 4/32 .... etc.
+ 1/16 = 2/32 .... etc.
+ 1/32 .... etc.
_____________________________ --- add each column up
= 1/4 3/8 7/16 15/32 .... etc.


what you'll notice is that the sums keep getting closer and closer to 1/2 but will never actually get there (if you graph this, it will be asymptotic to 1/2). you now have your upper-bound... 1/2

answer: 1/4 < result < 1/2, D

Great question. thanks.
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checking upto 5 terms will give the value.

(1/2-1/4)+(1/8-1/16)+(1/32-1/64)+ ....

1/4 + 1/16 + 1/64 + 1/128 + ..

value will be between 1/4 < x < 1/2.

D
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