5-Day Free Trial
5-day free, full-access trial TTP
Available with Beat the GMAT members only code
MORE DETAILS
Sequence
This topic has expert replies
-
GMATGuruNY
- GMAT Instructor
- Posts: 15539
- Joined: Tue May 25, 2010 12:04 pm
- Location: New York, NY
- Thanked: 13060 times
- Followed by:1906 members
- GMAT Score:790
Notice that the answer choices are RANGES.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
We are not expected to calculate the exact sum.
Use a NUMBER LINE to determine the correct range.
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.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
GMAT/MBA Expert
-
Brent@GMATPrepNow
- GMAT Instructor
- Posts: 16207
- Joined: Mon Dec 08, 2008 6:26 pm
- Location: Vancouver, BC
- Thanked: 5254 times
- Followed by:1268 members
- GMAT Score:770
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
Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
GMAT/MBA Expert
-
[email protected]
- Elite Legendary Member
- Posts: 10392
- Joined: Sun Jun 23, 2013 6:38 pm
- Location: Palo Alto, CA
- Thanked: 2867 times
- Followed by:511 members
- GMAT Score:800
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
GMAT assassins aren't born, they're made,
Rich
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
GMAT assassins aren't born, they're made,
Rich
-
Matt@VeritasPrep
- GMAT Instructor
- Posts: 2630
- Joined: Wed Sep 12, 2012 3:32 pm
- Location: East Bay all the way
- Thanked: 625 times
- Followed by:119 members
- GMAT Score:780
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.
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.
