Problem Solving

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

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.

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!

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.

Tatiana

(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.

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.