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For every integer k from 1 to 10 inclusive the kth term of a

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Manager
Joined: 13 Apr 2006
Posts: 56
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For every integer k from 1 to 10 inclusive the kth term of a [#permalink]  11 Jun 2006, 16:04
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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, 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

Please explain your answer. This to me was taking too long so I guessed and moved on. Is there a quick, efficient way to execute this?
VP
Joined: 29 Dec 2005
Posts: 1348
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Kudos [?]: 37 [0], given: 0

Re: GMAT Prep Algebra PS [#permalink]  11 Jun 2006, 17:07
kuristar wrote:
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, 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 guess -1^(k+1)*(1/2^k) = (-1)^(k+1)*(1/2^k)

1st term = (-1)^(1+1)*(1/2^k) = 1/2
2nd term = (-1)^(k+1)*(1/2^k) = -1/4
3rd term = (-1)^(k+1)*(1/2^k) = 1/8
4th term = (-1)^(k+1)*(1/2^k) = -1/16

now we can get every term by multiplying the prededing term by -1/2. so..

5th = 4th term (-1/2) = -1/16 (-1/2) = 1/32
6th = - 1/64
7th = 1/128
8th = -1/256
9th = 1/512
10th = -1/1024

get the sum of every 2 conseqcutive terms
sum of 1st and 2nd term = 1/2 - 1/4 = 1/4
sum of 3rd and 4th term = 1/8 - 1/16 = 1/16

similarly the next term becames = (1/16)(1/4)=1/64

the next term becames = (1/64)(1/4)=1/256

the final term becames = (1/256)(1/4)=1/1024

lets add them all =1/4 +1/16+1/64+1/256+1/1024 = 331/1024

so this value is ====>>>>>

1/4 < 331/1024 < 1/3

it is D.
SVP
Joined: 30 Mar 2006
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Kudos [?]: 46 [1] , given: 0

1
KUDOS
1st number of the sequence = 1/2
2nd number = -1/4
3rd number = 1/8
Now this is a geometric series with r = -1/2

Sum = a(1-r^n)/1-r
= 1/2(1- (-1/2)^10)/1-(-1/2)

= 2^10 -1/3*2^10
~<1/3
Hence D
Manager
Joined: 10 May 2006
Posts: 186
Location: USA
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Kudos [?]: 3 [0], given: 0

K1= 1/2
K2 = (-1/4)
K3 = 1/8
K4 = (-1/16)

and so forth...as K gets larger, the numbers get smaller and smaller.

Group the numbers together and add together
ie) K1+ K2 = 1/2 + -1/4 = 1/4
K3 +K4 = 1/8 + -1/16 = 1/8

as we continue this pattern, the figures will become smaller and smaller positive numbers.

Therefore we know that the answer is definitely greater than 1/4, but since the numbers get smaller and smaller, the best answer is D, between 1/4 and 1/2.
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