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by GMATGuruNY » Sat May 07, 2016 5:35 am
For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is give by (see attachment). 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
Notice that the answer choices are RANGES.
We are not expected to calculate the exact sum.
Use a NUMBER LINE to determine the correct range.

Image

Follow the arrows.
The first term is 1/2.
When we add in -1/4 -- the second term -- the sum decreases to 1/4.
When we add in +1/8 -- the third term -- the sum increases to 3/8.
By now, we can already see that the sum will converge to a value somewhere between 1/4 and 3/8.

The correct answer is D.
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by Brent@GMATPrepNow » Sat May 07, 2016 6:37 am
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

Plug in some values of k to see that 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

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by [email protected] » Sat May 07, 2016 8:56 am
Hi eitijan,

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 bit of work, you can correctly deduce what the sequence is:

+1/2, -1/4, +1/8, -1/16, etc.

The "key" to solving this question quickly is to think about the terms in "sets of 2"...

1/2 - 1/4 = 1/4

Since the first term in each "set 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

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by Matt@VeritasPrep » Thu May 12, 2016 12:34 am
And now for something completely different!

Suppose we could do this forever. We'd have

1/2 - 1/4 + 1/8 - 1/16 ± ... = x

or

1/4 + 1/16 + 1/64 + ... = x

or

1/4 * (1 + 1/4 + 1/16 + ...) = x

or

1/4 * (1 + x) = x

or

x = 1/3

Since we only have part of this sum of an infinite number of terms, we must be somewhere close to 1/3, or between 1/4 and 1/2.
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