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by uncbeers » Tue Jul 22, 2008 10:16 pm
Can anyone help me with this GMAT Prep question? I hate these ones so much...and it is probably a simple answer.

For every integer from k 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:

-greater than 2
-between 1 and 2
-between 1/2 and 1
-between 1/4 and 1/2
-less than 1/4


Thanks all!












Answer is......D
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by VP_Tatiana » Wed Jul 23, 2008 7:59 am
A good strategy for these types of questions is to write out the first few terms of the sequence to see if a pattern emerges. We have:

(-1)^k+1 (1/2^k).

I'm a little unclear whether you mean:

((-1)^k+1 )(1/2^k) or -1^(k+1 (1/2^k)). I am going to solve this with the assumption that you maen the former.

k=1
(-1^2)(1/2) = 1/2

k=2
(-1^3)(1/4) = -1/4

... I can see that the terms oscillate between negative and positive, and that the magnitude of the term is cut in half each time. So, we start with 1/2, then subtract 1/4, then add and subtract fractions smaller than 1/4. Thus, the answer is between 1/4 and 1/2.
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by uncbeers » Wed Jul 23, 2008 8:33 am
I should have added one more set of parantheses. You have it right the way you went about it. It makes perfect sense and now I feel dumb. I find some of these questions written so awkwardly it throws me for a loop! :?

Thank you for doing that though-seems very clear now!
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by Ian Stewart » Thu Jul 24, 2008 8:45 am
VP_Tatiana wrote: So, we start with 1/2, then subtract 1/4, then add and subtract fractions smaller than 1/4. Thus, the answer is between 1/4 and 1/2.
That isn't quite right, logically. Just because you add fractions less than a quarter doesn't guarantee the sum is less than 1/2. For example, 1/4 + 1/8 + 1/8 + 1/8 is greater than 1/2. It's because we are adding fractions which are much smaller than 1/4 that we can be sure the sum is not larger than 1/2. We have:

1/2 - 1/4 + 1/8 - 1/16 + 1/32 - ... - 1/2048 =

1/4 + 1/16 + 1/64 + 1/256 + 1/1024

Now you can see by estimation that the sum is less than 1/2, or you can see that the sum is smaller than 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ... which will never reach 1/2 no matter how many terms you add.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by VP_Tatiana » Thu Jul 24, 2008 9:07 am
Hi Ian,

I explained that these were alternating signs and 1/2 the magnitude each time, so the example of 1/8 + 1/8 +1/8 doesn't parallel what is going on in this particular problem. The fact that each term is the opposite sign from the last, and half the magnitude of the last, is what insures the answer will stay bounded between 1/4 and 1/2.

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by mkingsly » Tue Jul 29, 2008 11:24 am
(1/2^1) - (1/2^2) + (1/2^3) - ... - (1/2^10).
This is a geometric series with a= 1/2, r= -1/2, n= 10. By substituting these values in
T = a(1- r^(n+1))/(1-r),
we get 1/3 (approx), which is between 1/4 and 1/2.
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by II » Sat Aug 23, 2008 8:49 am
As already pointed out this is a geometric sequence.

For sure it helps to write out a few terms to identify the pattern and classify the sequence:

If the new term is formed by adding a fixed amount to the previous term (common difference), then this is an arithmetic sequence.

If the new term is formed by multiplying the previous term with a fixed value (common ratio), then this is a geometric sequence.

In this case we are dealing with a geometric sequence ...
The common ratio in this case -(1/2)

See attached some additional details on arithmetic and geometric sequences. Especially the useful formulas to apply for each scenario.
Attachments
Arithmetic Progression Formulas and Notes.jpg
Arithmetic Progression Fomula/Notes
Geometric Progression Formulas and Notes.jpg
Geometric Progression Fomula/Notes
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