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# sum of sequence

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sum of sequence [#permalink]  08 Oct 2009, 10:45
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100% (00:00) correct 0% (00:00) wrong based on 1 sessions
I thought it tapered off inbetween 1/2 and 1 but I think the wording kinda confused me

Thanks for the help
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Re: sum of sequence [#permalink]  08 Oct 2009, 11:13
keeshafey wrote:
I thought it tapered off inbetween 1/2 and 1 but I think the wording kinda confused me

Thanks for the help

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.
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Re: sum of sequence [#permalink]  08 Oct 2009, 11:16
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I had the same question in gmatprep.
1/2 - 1/4 + 1/8 - 1/16 + 1/32 + ....
lets keep first 1/2 on the side.
Now you have
-1/4 + 1/8 = -1/4 negative
-1/16 + 1/32 = -1/16 negative
.... = negative
Here negative term is twice as much as positive terms
-2x+x= -x
so basically you are doing
1/2 - x1- x2 -x3 -x4...
so answer will be always less than 1/2

Another way to look at it...I used this way but it too k me long time..
Basically you have this right
1/2 - 1/4 + 1/8 - 1/16 + 1/32 + ....

just take the two terms at a time...
1/2 - 1/4 = 1/4
1/8-1/16=1/16
1/32-1/64=1/64
Basically each positive term is double than negative term so you are doing
2x-x=x

now you have
1/4+1/16+1/64+1/256+1/1024
1/4 = 0.25
1/16 = 0. 0625 and each subsequent term is 1/4 of 0.0625 which cannot add upto another 0.25 so answer has to be less than 1/2

dont forget to give kudos if you like it.
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Re: sum of sequence [#permalink]  08 Oct 2009, 14:29
Expert's post
orsang8 wrote:
I had the same question in gmatprep.
1/2 - 1/4 + 1/8 - 1/16 + 1/32 + ....
lets keep first 1/2 on the side.
Now you have
-1/4 + 1/8 = -1/4 negative
-1/16 + 1/32 = -1/16 negative
.... = negative
Here negative term is twice as much as positive terms
-2x+x= -x
so basically you are doing
1/2 - x1- x2 -x3 -x4...
so answer will be always less than 1/2

Another way to look at it...I used this way but it too k me long time..
Basically you have this right
1/2 - 1/4 + 1/8 - 1/16 + 1/32 + ....

just take the two terms at a time...
1/2 - 1/4 = 1/4
1/8-1/16=1/16
1/32-1/64=1/64
Basically each positive term is double than negative term so you are doing
2x-x=x

now you have
1/4+1/16+1/64+1/256+1/1024
1/4 = 0.25
1/16 = 0. 0625 and each subsequent term is 1/4 of 0.0625 which cannot add upto another 0.25 so answer has to be less than 1/2

dont forget to give kudos if you like it.

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

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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Re: sum of sequence [#permalink]  08 Oct 2009, 17:01
thanks everyone. I had done the same thing but I guess I missed something in the beginning to make me think it was more than 1/2
Re: sum of sequence   [#permalink] 08 Oct 2009, 17:01
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