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PS- integers and number properties [#permalink]
07 Jan 2006, 16:43

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Hey, here's another one from the gmat prep. I do not understand the question. Please help!

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, what is the value range for T?

note: -1 is raised to k+1, and the 2 of 1/2 is raised k

k = 1 to 10
So,
1st term of the series = (-1)^2 x (1/2) = 1/2
2nd term of the series = (-1)^3 x (1/4) = -1/4
3rd term of the series = (-1)^4 x (1/8) = 1/8

Hence, common ratio r = -1/2 --> Geometric series
T is the sum of k terms and r < 1, so:
T = a[(1-r^k)/(1-r)] = 0.5[(1-(-0.5)^10)/91+0.5)] = 1023/(1024*3)
T ~= 1/3 _________________

1/3 is between 1/4 and 1/2.
I vaguely remember this question and from the answer choices, I think the only one that accomodates 1/3 is "between 1/4 and 1/2".

Guys, the question asks for the value range for T.

This confused me at the start ..... value range = sum ??!?! I wouldve just calculated the first term and the tenth term, and then subtracted the two to find the 'value range' ....

How in the world do you guys figure this out? Can some one please explain this...

Hence, common ratio r = -1/2 --> Geometric series
T is the sum of k terms and r < 1, so:
T = a[(1-r^k)/(1-r)] = 0.5[(1-(-0.5)^10)/91+0.5)] = 1023/(1024*3)
T ~= 1/3

If the terms in a series have a common ratio, its a geometric series and there are generalised formulae for working out the nth term of the series and the sum of n terms.

If the terms have a common difference its an arithmetic series and again there are formulae for the nth term and sum of n terms.

So in the question above, I've worked out the common ratio (from the terms) and then used the formula for sum to get the value for T.

How in the world do you guys figure this out? Can some one please explain this...

Hence, common ratio r = -1/2 --> Geometric series T is the sum of k terms and r < 1, so: T = a[(1-r^k)/(1-r)] = 0.5[(1-(-0.5)^10)/91+0.5)] = 1023/(1024*3) T ~= 1/3

Thanks!

Can someone please explain the part highlighted.

where did 91 come from , should (1-r) = 1 - (-0.5)
I know I am missing an important point here.

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