Hi nchaswal,
This question comes up every so often in this Forum. There are a couple of different ways of thinking about this problem, but they all require a certain degree of "math."
With a little bit of work, you can determine the sequence of numbers...
+1/2, -1/4, +1/8, -1/16, etc.
The "key" to solving this question quickly is to think about the terms in "pairs"...
1/2 - 1/4 = 1/4
Since the first term in each "pair of 2" is greater than the second (negative) term, we now know that each set of 2 will be positive.
1/8 - 1/16 = 1/16
Now we know that each additional set of 2 will be significantly smaller than the prior set of 2.
Without doing all of the calculations, we know....
We have 1/4 and we'll be adding tinier and tinier fractions to it. Since there are only 10 terms in the sequence, there are only 5 sets of 2, so we won't be adding much to 1/4. Based on the answer choices, only one answer makes any sense...
Final Answer: D
GMAT assassins aren't born, they're made,
Rich
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For every integer k from 1 to 10.....
Source: Beat The GMAT — Problem Solving |
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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
T = 1/2 - 1/4 + 1/8 - 1/16 + . . .
Notice that we can rewrite this as T = (1/2 - 1/4) + (1/8 - 1/16) + . . .
When you start simplifying each part in brackets, you'll see a pattern emerge. We get...
T = 1/4 + 1/16 + 1/64 + 1/256 + 1/1024
Now examine the last 4 terms: 1/16 + 1/64 + 1/256 + 1/1024
Notice that 1/64, 1/256, and 1/1024 are each less than 1/16
So, (1/16 + 1/64 + 1/256 + 1/1024) < (1/16 + 1/16 + 1/16 + 1/16)
Note: 1/16 + 1/16 + 1/16 + 1/16 = 1/4
So, we can conclude that 1/16 + 1/64 + 1/256 + 1/1024 = (a number less than 1/4)
Now start from the beginning: T = 1/4 + (1/16 + 1/64 + 1/256 + 1/1024)
= 1/4 + (a number less 1/4)
= A number less than 1/2
Of course, we can also see that T > 1/4
So, [spoiler]1/4 < T < 1/2[/spoiler]
Answer: D
Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
Thanks Rich. I did find the pattern but in a longer way.Brent@GMATPrepNow wrote: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
Of course, we can also see that T > 1/4
So, [spoiler]1/4 < T < 1/2[/spoiler]
Answer: D
Cheers,
Brent
Thanks Brent. That way is really quite different. I could not think that way. Surely helps.
It is GMAT. So what?
