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# PS question from Gmat Prep. software

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Manager
Joined: 12 Jun 2006
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PS question from Gmat Prep. software [#permalink]  14 Nov 2006, 15:55
00:00

Difficulty:

5% (low)

Question Stats:

40% (07:40) correct 60% (01:23) wrong based on 6 sessions
Can someone help me solve the below question?

Q. For every integer K from 1 to 10, inclusive, the Kth term of a certain sequence is given by (-1)^K+1 . (1/2^K). If T is the sum of the first 10 terms in the sequence, then T is

a) > 2
b) between 1 and 2
c) between 1/2 and 1
d) between 1/4 and 1/2
e) less then 1/4

Thanks
GMAT Club Legend
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(-1)^K+1 . (1/2^K). --> The dots mean multiply?
Manager
Joined: 12 Jun 2006
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ywilfred wrote:
(-1)^K+1 . (1/2^K). --> The dots mean multiply?

Yes that is correct
Senior Manager
Joined: 01 Sep 2006
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Q. For every integer K from 1 to 10, inclusive, the Kth term of a certain sequence is given by (-1)^K+1 . (1/2^K). If T is the sum of the first 10 terms in the sequence, then T is

a) > 2
b) between 1 and 2
c) between 1/2 and 1
d) between 1/4 and 1/2
e) less then 1/4

T1 = (-1)^2*(1/2)^1= 1/2
T2= (-1)^3*(1/2)^2 = -1/4
T3= (-1)^4*(1/2)^3 = + 1/8
.
.
T9= (-1)^10*(1/2)^9= (1/2)^9
t10=(-1)^11*(1/2)^10= - (1/2)^10

ans b/w 1/2 and 1

i think this problem is based on using the knowlege that fractions when multiplied by a fraction the value of product reduces

so 1/2 multiped by 1/2 is less than 1/2
and so on

since 1st term is 1/2 the sume should be greatd than 1/2

if u add rest of the trms the sume will b less than 1/2
ANS B between 1/2 and 1
Manager
Joined: 10 Jul 2006
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I got D for this one. I used the same logic as Damager did. However, I think since the sequence alters a positive and a negative, ie the first term is 1/2, the second is -1/4, the third is 1/8, fourth is -1/16. So even with the first two terms, the sum should be 1/4 and then adding smaller and smaller amount as the sequence goes. So Asn D between 1/4 and 1/2
Manager
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enola wrote:
I got D for this one. I used the same logic as Damager did. However, I think since the sequence alters a positive and a negative, ie the first term is 1/2, the second is -1/4, the third is 1/8, fourth is -1/16. So even with the first two terms, the sum should be 1/4 and then adding smaller and smaller amount as the sequence goes. So Asn D between 1/4 and 1/2

So adding smaller and smaller amounts makes it go to 1/2 even though the terms are positive and negative. Can you please explain a bit more about your logic?

Thanks
Intern
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--------------------------------------------------------------------------------

Q. For every integer K from 1 to 10, inclusive, the Kth term of a certain sequence is given by (-1)^K+1 . (1/2^K). If T is the sum of the first 10 terms in the sequence, then T is

a) > 2
b) between 1 and 2
c) between 1/2 and 1
d) between 1/4 and 1/2
e) less then 1/4

T1 = (-1)^2*(1/2)^1= 1/2
T2= (-1)^3*(1/2)^2 = -1/4
T3= (-1)^4*(1/2)^3 = + 1/8
.
.
T9= (-1)^10*(1/2)^9= (1/2)^9
t10=(-1)^11*(1/2)^10= - (1/2)^10

1/2 - 1/4 = 1/4 + 1/8 = 3/8 - 1/16 = 5/16 + 1/32 = 11 /32 - 1/64 = 21 / 64 + 1/128 = 43 / 128

since you are summing, the number is going to slowly move between 1/4 and 1/2 but staying above 1/4 and below 1/2. It keeps moving by smaller and smaller fractions, but it always will be above 1/4 and under 1/2. Look at it this way. The first two numbers establish the range. It starts at a 1/2 then drops to 1/4 and each time it lowers and raises by a smaller increment. What you will have is something that looks like this:

..
....
.......
........
...........
.............
.................
.............
...........
.........
.......
....
..

meaning, first the number swings a lot, then it slowly swings by increasingly an infinitely smaller increments. Eventually you are adding 1/10000000 and then subtracting 1/100000000 and so on into infinity.

Long story short, you have the number .5 and .25. The number made by this sum will just keep getting more and more decimal places instead of moving past one of those two numbers.

I really hope this long ass explanation is right or it will look pretty stupid .

I'm sitting in Norway and its almost 1am, so who knows.
SVP
Joined: 08 Nov 2006
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The series 1/2-1/4+1/8-1/16...is actually a geometric progression with the first term 1/2 and every subsequent term multiplied by -1/2.

a,ar,ar2,.....etc

Sum of 1st n terms of GP = a(1-r^[n+1])/(1-r)

Sum of 10 terms = 1/2(1-(-1/2)^[10+1])/(1-(-1/2) = 0.33

Answer : Between 1/4 and 1/2.

Hope this helps.
Senior Manager
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Re: [#permalink]  24 Jul 2011, 17:39
ncp wrote:
The series 1/2-1/4+1/8-1/16...is actually a geometric progression with the first term 1/2 and every subsequent term multiplied by -1/2.

a,ar,ar2,.....etc

Sum of 1st n terms of GP = a(1-r^[n+1])/(1-r)

Sum of 10 terms = 1/2(1-(-1/2)^[10+1])/(1-(-1/2) = 0.33

Answer : Between 1/4 and 1/2.

Hope this helps.

Isnt the forumula for GP a(1-r^n)/1-r? See youtube video here : http://www.youtube.com/watch?v=OkPI1_BKo9w
Intern
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Re: PS question from Gmat Prep. software [#permalink]  25 Jul 2011, 03:30
ncp, what is short way to calculate (1+1/2^11) ? there is gotta be a shortcut..rather than calculating 2^11..
Re: PS question from Gmat Prep. software   [#permalink] 25 Jul 2011, 03:30
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