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Sequence Again
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Calculate until you see the pattern.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 first 10 terms in the sequence, then T is
A. Greater than 2
B. Between 1 and 2
C. Between 1/2 to 1
D. Between 1/4 to1/2
E. Less than ¼
Some test-takers might find it helpful to visualize the sum on a number line.
If k=1, -1^(1+1)*(1/2*1) = 1/2
If k=2, -1^(2+1)*(1/2*2) = -1/4
Sum of the first two terms is 1/2 + ( -1/4) = 1/4.
If k=3, -1^(3+1)*(1/2*3) = 1/8.
If k=4, -1^(4+1)*(1/2*4) = -1/16
Now we can see the pattern.
The sum increases by a fraction (1/8, for example) and then decreases by a fraction 1/2 the size (1/16).
In other words, the sum will alternate between increasing a little and then decreasing a little less than it went up.
The sum of the first 2 terms is 1/4. From there, the sum will increase by 1/8, decrease by a smaller fraction (1/16), increase by an even smaller fraction (1/32), and so on. Here are the first four terms, plotted on a number line:
Since all of the fractions after the first two terms will be less than 1/4, the sum will end up somewhere between 1/4 and 1/2.
The correct answer is D.
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Hi shibsriz,
This is a layered sequence question that would take a decent chunk of time to answer IF you did the entire calculation. Oftentimes, when a "math approach" takes too long, the question will be built on some type of pattern that can be discovered. You'll still have to do some work, but it won't be nearly as much as if you calculated the answer.
Here are the clues that will help us save some time:
1) The answer choices are RANGES, which means that we don't necessarily need to know the exact answer.
2) We're only dealing with 10 terms
3) (-1)^(K+1) will go "back and forth" from positive to negative...
When K=1, we get a positive
When K=2, we get a negative
When K=3, we get a positive
When K=4, we get a negative
etc.
Adding up these 10 terms will actually lead to some subtraction (all of the negative terms)
4) (1/2^K) will get smaller and smaller as K gets bigger
With this info, here's the logic that I'd use:
The first term (K=1) gives us +1/2
The second term (K=2) gives us -1/4
We'll end up adding 4 more positive fractions (that get smaller and smaller) and 4 more negative fractions (which reduce the total).
Since 1/2 - 1/4 = 1/4 and we'll just be dealing with some more tiny fractions (some added, some subtracted), the total would have to be a little more than 1/4.
Final Answer: D
GMAT assassins aren't born, they're made,
Rich
This is a layered sequence question that would take a decent chunk of time to answer IF you did the entire calculation. Oftentimes, when a "math approach" takes too long, the question will be built on some type of pattern that can be discovered. You'll still have to do some work, but it won't be nearly as much as if you calculated the answer.
Here are the clues that will help us save some time:
1) The answer choices are RANGES, which means that we don't necessarily need to know the exact answer.
2) We're only dealing with 10 terms
3) (-1)^(K+1) will go "back and forth" from positive to negative...
When K=1, we get a positive
When K=2, we get a negative
When K=3, we get a positive
When K=4, we get a negative
etc.
Adding up these 10 terms will actually lead to some subtraction (all of the negative terms)
4) (1/2^K) will get smaller and smaller as K gets bigger
With this info, here's the logic that I'd use:
The first term (K=1) gives us +1/2
The second term (K=2) gives us -1/4
We'll end up adding 4 more positive fractions (that get smaller and smaller) and 4 more negative fractions (which reduce the total).
Since 1/2 - 1/4 = 1/4 and we'll just be dealing with some more tiny fractions (some added, some subtracted), the total would have to be a little more than 1/4.
Final Answer: D
GMAT assassins aren't born, they're made,
Rich
