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

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Senior Manager
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For every integer k from 1 to 10, inclusive, the kth term of a certain [#permalink] New post 11 Aug 2007, 12:14
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C
D
E

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Question Stats:

56% (03:10) correct 44% (01:31) wrong based on 51 sessions
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. greater than 2

B. between 1 and 2

C. between 0.5 and 1

D. between 0.25 and 0.5

E. less than 0.25


OPEN DISCUSSION OF THIS QUESTION IS HERE: for-every-integer-k-from-1-to-10-inclusive-the-kth-term-of-88874.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 19 Dec 2014, 07:08, edited 1 time in total.
Renamed the topic, edited the question, added the OA and moved to PS forum.
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Re: For every integer k from 1 to 10, inclusive, the kth term of a certain [#permalink] New post 11 Aug 2007, 12:22
zakk wrote:
1) 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 of the sequence, then T is:

a) greater than 2
b) between 1 and 2
c) between ½ and 1
d) between ¼ and ½
e) less than ¼


I guess right, but have no idea on the concept behind this.


Can you clarify (-1)^k+1 (1/2^k)?
Is it ((-1)^(k+1)) * (1/(2^K))?
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Re: For every integer k from 1 to 10, inclusive, the kth term of a certain [#permalink] New post 11 Aug 2007, 12:36
i interpreted (-1)^k+1 (1/2^k) as --> ((-1)^(k+1)) * (1/(2^K))

i believe the answer is D.

the first term will always be either 1 or -1, the second term will be 1/2 for k=1, then 1/2 squared or 1/4 for k=2, then 1/2 cubed, etc.

so the first ten terms will be:

1/2 - 1/4 + 1/8 - 1/16 + 1/32... - 1/1024

the values decrease with each term, so the sum of the terms is never greater than 1/2 or less than 1/4.
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Re: For every integer k from 1 to 10, inclusive, the kth term of a certain [#permalink] New post 11 Aug 2007, 13:52
I get B..

basically the first term -(1)^k gets cancelled out..we have 5 even number, 5 odd numbers from 1 thru 10 inclusive...

so then basically we are adding 1+ 1/2 + 1/4 +1/8....

this sum approcaches 1..however the sum is definetly between 1 and 2..
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Re: For every integer k from 1 to 10, inclusive, the kth term of a certain [#permalink] New post 11 Aug 2007, 13:56
assuming its the later case -1^(K+1)

then its basically saying that for every odd K, the term would be positive and even term would be negative, however net result would they still cancel out...in that case the sum would be 1/2+1/4+1/8...i.e C

bkk145 wrote:
zakk wrote:
1) 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 of the sequence, then T is:

a) greater than 2
b) between 1 and 2
c) between ½ and 1
d) between ¼ and ½
e) less than ¼


I guess right, but have no idea on the concept behind this.


Can you clarify (-1)^k+1 (1/2^k)?
Is it ((-1)^(k+1)) * (1/(2^K))?
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Re: For every integer k from 1 to 10, inclusive, the kth term of a certain [#permalink] New post 11 Aug 2007, 14:16
I agree with emarinich - and that's a great explanation, btw!

The first nmber in the sequence is 1/2, then -1/4, 1/8, etc. It increases or decreases by an exponentially smaller number each time, so matter how far you carry it, the sum will never equal a number greater than the first number, so must be between 1/4 and 1/2. Actually this same problem was posted a week or so ago - challenging and fascinating!
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Re: For every integer k from 1 to 10, inclusive, the kth term of a certain [#permalink] New post 11 Aug 2007, 14:50
emarinich wrote:
i interpreted (-1)^k+1 (1/2^k) as --> ((-1)^(k+1)) * (1/(2^K))

i believe the answer is D.

the first term will always be either 1 or -1, the second term will be 1/2 for k=1, then 1/2 squared or 1/4 for k=2, then 1/2 cubed, etc.

so the first ten terms will be:

1/2 - 1/4 + 1/8 - 1/16 + 1/32... - 1/1024

the values decrease with each term, so the sum of the terms is never greater than 1/2 or less than 1/4.


that is what the question was asking. sorry if it wasn't clear. Should have hotlinked it!

thanks for the explanation as well.
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Re: For every integer k from 1 to 10, inclusive, the kth term of a certain [#permalink] New post 19 Dec 2014, 05:21
emarinich wrote:
i interpreted (-1)^k+1 (1/2^k) as --> ((-1)^(k+1)) * (1/(2^K))

i believe the answer is D.

the first term will always be either 1 or -1, the second term will be 1/2 for k=1, then 1/2 squared or 1/4 for k=2, then 1/2 cubed, etc.

so the first ten terms will be:

1/2 - 1/4 + 1/8 - 1/16 + 1/32... - 1/1024

the values decrease with each term, so the sum of the terms is never greater than 1/2 or less than 1/4.



How can this tell that it cannot be less than 1/4??
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Re: For every integer k from 1 to 10, inclusive, the kth term of a certain [#permalink] New post 19 Dec 2014, 07:10
Expert's post
LUC999 wrote:
emarinich wrote:
i interpreted (-1)^k+1 (1/2^k) as --> ((-1)^(k+1)) * (1/(2^K))

i believe the answer is D.

the first term will always be either 1 or -1, the second term will be 1/2 for k=1, then 1/2 squared or 1/4 for k=2, then 1/2 cubed, etc.

so the first ten terms will be:

1/2 - 1/4 + 1/8 - 1/16 + 1/32... - 1/1024

the values decrease with each term, so the sum of the terms is never greater than 1/2 or less than 1/4.



How can this tell that it cannot be less than 1/4??


CHECK HERE: for-every-integer-k-from-1-to-10-inclusive-the-kth-term-of-88874.html
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Re: For every integer k from 1 to 10, inclusive, the kth term of a certain   [#permalink] 19 Dec 2014, 07:10
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