For every integer k from 1 to 10...

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NYC493: Junior | Next Rank: 30 Posts; Posts: 25; Joined: Tue Apr 05, 2011 5:45 amFollowed by: 1 member; Location: New York City

by NYC493 » Mon Mar 05, 2012 2:22 pm

Can someone tell me a quick way to decode a problem like this? If you start to solve each number in the sequence manually, you can see that it's alternating from positive to negative. But, how do you arrive at an approximate sum quickly?

"For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is given by (-1)^(k+1) x (1/2^k). If "T" is the sum of the first 10 terms in the sequence, then T is:

A. greater than 2
B. between 1 and 2
C. between 1/2 and 1
D. between 1/4 and 1/2
E. less than 1/2

Thanks!

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Source: Beat The GMAT — Problem Solving | Mon Mar 05, 2012 2:22 pm

pemdas: Legendary Member; Posts: 1084; Joined: Fri Apr 15, 2011 2:33 pmFollowed by: 21 members; Thanked: 158 times

by pemdas » Mon Mar 05, 2012 2:42 pm

NYC493 wrote:Can someone tell me a quick way to decode a problem like this? If you start to solve each number in the sequence manually, you can see that it's alternating from positive to negative. But, how do you arrive at an approximate sum quickly?

"For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is given by (-1)^(k+1) x (1/2^k). If "T" is the sum of the first 10 terms in the sequence, then T is:

A. greater than 2
B. between 1 and 2
C. between 1/2 and 1
D. between 1/4 and 1/2
E. less than 1/2

Thanks!

you can find many interesting solutions on BTG
i'll offer mine and will try to renew this question's thread

you are right the sign alternates and we get something like 1/2 -1/4 +1/8 -1/16 ... or (2^9 -2^8 +2^7 -2^6 +2^5 -2^4 ...)/2^10 by further simplification we get [2^8(2-1) +2^6(2-1) +2^4(2-1) ...)/2^10= (2^8 + 2^6 + 2^4 + 2^2)/2^10 = 2^2(2^6 + 2^4 +2^2 +1)/2^10 = (2^6 + 2^4 +2^2 +1)/2^8 or (64+16+4+1)/(16*16) [spoiler]2^8=(2^4)*(2^4)=16*16[/spoiler], 85/256 =~1/3

d

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krusta80: Master | Next Rank: 500 Posts; Posts: 143; Joined: Mon Mar 14, 2011 3:13 amFollowed by: 5 members; Thanked: 34 times

by krusta80 » Mon Mar 05, 2012 3:25 pm

NYC493 wrote:Can someone tell me a quick way to decode a problem like this? If you start to solve each number in the sequence manually, you can see that it's alternating from positive to negative. But, how do you arrive at an approximate sum quickly?

"For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is given by (-1)^(k+1) x (1/2^k). If "T" is the sum of the first 10 terms in the sequence, then T is:

A. greater than 2
B. between 1 and 2
C. between 1/2 and 1
D. between 1/4 and 1/2
E. less than 1/2

Thanks!

Pemdas's method is very impressive...a lot can be learned from the way he manipulated the algebra. That said, though, during a test -- when time is of the essence -- I would probably do something along the following lines...

STEP 1

Formulas of these types can be generalized as...

(-1)^(i) * (r)^j

The first thing I look at is the value of r. Specifically, is r less than 1 or greater than 1? If it's less than 1 (btw...it almost always will be in my experience), then I know that the sum will change less and less as j increases. This is because a fraction less than 1 will get smaller and smaller as it's exponent increases.

The next thing to do is to figure out how many terms are enough to determine the right choice. I do this by first finding the smallest range from the given choices. In this case...

A) Range is infinite
B) Range is 1
C) Range is 1/2
D) Range is 1/4
E) Range is 1/2

***Side note: The GMAT is designed to challenge us...to trick us at times. Therefore, right away I'm leaning toward either C,D, or E. Why? Because their target ranges are the smallest, and the smaller the range, the more terms required to determine whether that answer choice is correct.

So, the smallest range is 1/4. What power of 1/2 gives us 1/4? 2 of course. This means that it'll take two terms to reach that range's level of precision. For this problem:

1/2 - 1/4 = 1/4 OK, so we're close already. It's either going to be D or E.

Once you reach the required level of precision, I would be safe by going with two more terms (notice always an EVEN number of terms to cancel out the opposing signs):

1/4 + 1/8 - 1/16 = (4+2-1)/16 = 5/16.

This leaves choice D. Hey what do you know...they picked the narrowest range!

Last edited by krusta80 on Mon Mar 05, 2012 6:37 pm, edited 1 time in total.

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NYC493: Junior | Next Rank: 30 Posts; Posts: 25; Joined: Tue Apr 05, 2011 5:45 amFollowed by: 1 member; Location: New York City

by NYC493 » Mon Mar 05, 2012 4:37 pm

Thank you both! I'll have to keep practicing problems of this type. But, your strategies are appreciated!

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Mon Mar 05, 2012 7:05 pm

NYC493 wrote:Can someone tell me a quick way to decode a problem like this? If you start to solve each number in the sequence manually, you can see that it's alternating from positive to negative. But, how do you arrive at an approximate sum quickly?

"For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is given by (-1)^(k+1) x (1/2^k). If "T" is the sum of the first 10 terms in the sequence, then T is:

A. greater than 2
B. between 1 and 2
C. between 1/2 and 1
D. between 1/4 and 1/2
E. less than 1/2

Thanks!

We know that the sum of n terms of a geometric series is given by:
S(n) = a(1 - râ�¿)(1 - r), where a is the first term, r is the common ratio of the geometric progression and n = number of terms.

Here, a = 1/2, r = -1/2, n = 10
T = 1/2[1 - (-1/2)^10]/[1 + 1/2]
= 1/2[1 - 1/1024]/[3/2]
= 1/2 * 1023/1024 * (2/3)
= (1023/1024) * (1/3)
Now (1023/1024) = 1 approx, so T = 1/3, which lies between 1/4 and 1/2.

The correct answer is D.

Anurag Mairal, Ph.D., MBA
GMAT Expert, Admissions and Career Guidance
Gurome, Inc.
1-800-566-4043 (USA)

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aircraft: Newbie | Next Rank: 10 Posts; Posts: 6; Joined: Wed May 15, 2013 3:12 pm

by aircraft » Mon May 20, 2013 3:34 pm

Excuse me...could it be possible that Mr Mairal means S(n) = a(1 - râ�¿)/(1 - r) instead of S(n) = a(1 - râ�¿)(1 - r).

Thanks!

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Mon May 20, 2013 9:58 pm

NYC493 wrote:Can someone tell me a quick way to decode a problem like this? If you start to solve each number in the sequence manually, you can see that it's alternating from positive to negative. But, how do you arrive at an approximate sum quickly?

"For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is given by (-1)^(k+1) x (1/2^k). If "T" is the sum of the first 10 terms in the sequence, then T is:

A. greater than 2
B. between 1 and 2
C. between 1/2 and 1
D. between 1/4 and 1/2
E. less than 1/2

Thanks!

Another option is to get a better idea of this sum, by finding a few terms:

T = 1/2 - 1/4 + 1/8 - 1/16 + . . .
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 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

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gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pmFollowed by: 8 members; Thanked: 11 times

by gmattesttaker2 » Fri Jul 26, 2013 6:21 pm

Brent@GMATPrepNow wrote:
NYC493 wrote:Can someone tell me a quick way to decode a problem like this? If you start to solve each number in the sequence manually, you can see that it's alternating from positive to negative. But, how do you arrive at an approximate sum quickly?

"For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is given by (-1)^(k+1) x (1/2^k). If "T" is the sum of the first 10 terms in the sequence, then T is:

A. greater than 2
B. between 1 and 2
C. between 1/2 and 1
D. between 1/4 and 1/2
E. less than 1/2

Thanks!
Another option is to get a better idea of this sum, by finding a few terms:

T = 1/2 - 1/4 + 1/8 - 1/16 + . . .
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 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

Hello Brent,

Thanks a lot for the excellent explanation. I was clear till this point:

Of course, we can also see that T > 1/4

Is there a way we can arrive at this result similar to how you have concluded that T = A number less than 1/2

Thanks for your help.

Best Regards,
Sri

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gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pmFollowed by: 8 members; Thanked: 11 times

by gmattesttaker2 » Fri Jul 26, 2013 6:22 pm

Brent@GMATPrepNow wrote:
NYC493 wrote:Can someone tell me a quick way to decode a problem like this? If you start to solve each number in the sequence manually, you can see that it's alternating from positive to negative. But, how do you arrive at an approximate sum quickly?

"For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is given by (-1)^(k+1) x (1/2^k). If "T" is the sum of the first 10 terms in the sequence, then T is:

A. greater than 2
B. between 1 and 2
C. between 1/2 and 1
D. between 1/4 and 1/2
E. less than 1/2

Thanks!
Another option is to get a better idea of this sum, by finding a few terms:

T = 1/2 - 1/4 + 1/8 - 1/16 + . . .
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 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

Hello Brent,

Thanks a lot for the excellent explanation. I was clear till this point:

Of course, we can also see that T > 1/4

Is there a way we can arrive at this result similar to how you have concluded that T = A number less than 1/2

Thanks for your help.

Best Regards,
Sri

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gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pmFollowed by: 8 members; Thanked: 11 times

by gmattesttaker2 » Fri Jul 26, 2013 6:23 pm

Brent@GMATPrepNow wrote:
NYC493 wrote:Can someone tell me a quick way to decode a problem like this? If you start to solve each number in the sequence manually, you can see that it's alternating from positive to negative. But, how do you arrive at an approximate sum quickly?

"For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is given by (-1)^(k+1) x (1/2^k). If "T" is the sum of the first 10 terms in the sequence, then T is:

A. greater than 2
B. between 1 and 2
C. between 1/2 and 1
D. between 1/4 and 1/2
E. less than 1/2

Thanks!
Another option is to get a better idea of this sum, by finding a few terms:

T = 1/2 - 1/4 + 1/8 - 1/16 + . . .
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 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

Hello Brent,

Thanks a lot for the excellent explanation. I was clear till this point:

Of course, we can also see that T > 1/4

Is there a way we can arrive at this result similar to how you have concluded that T = A number less than 1/2

Thanks for your help.

Best Regards,
Sri

Like this post

gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pmFollowed by: 8 members; Thanked: 11 times

by gmattesttaker2 » Fri Jul 26, 2013 6:53 pm

Brent@GMATPrepNow wrote:
NYC493 wrote:Can someone tell me a quick way to decode a problem like this? If you start to solve each number in the sequence manually, you can see that it's alternating from positive to negative. But, how do you arrive at an approximate sum quickly?

"For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is given by (-1)^(k+1) x (1/2^k). If "T" is the sum of the first 10 terms in the sequence, then T is:

A. greater than 2
B. between 1 and 2
C. between 1/2 and 1
D. between 1/4 and 1/2
E. less than 1/2

Thanks!
Another option is to get a better idea of this sum, by finding a few terms:

T = 1/2 - 1/4 + 1/8 - 1/16 + . . .
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 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

Hello Brent,

Thanks a lot for the excellent explanation. I was clear till this point:

Of course, we can also see that T > 1/4

Is there a way we can arrive at this result similar to how you have concluded that T = A number less than 1/2

Thanks for your help.

Best Regards,
Sri

Like this post

gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pmFollowed by: 8 members; Thanked: 11 times

by gmattesttaker2 » Fri Jul 26, 2013 6:58 pm

Brent@GMATPrepNow wrote:
NYC493 wrote:Can someone tell me a quick way to decode a problem like this? If you start to solve each number in the sequence manually, you can see that it's alternating from positive to negative. But, how do you arrive at an approximate sum quickly?

"For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is given by (-1)^(k+1) x (1/2^k). If "T" is the sum of the first 10 terms in the sequence, then T is:

A. greater than 2
B. between 1 and 2
C. between 1/2 and 1
D. between 1/4 and 1/2
E. less than 1/2

Thanks!
Another option is to get a better idea of this sum, by finding a few terms:

T = 1/2 - 1/4 + 1/8 - 1/16 + . . .
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 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

Hello Brent,

Thanks a lot for the excellent explanation. I was clear till this point:

Of course, we can also see that T > 1/4

Is there a way we can arrive at this result similar to how you have concluded that T = A number less than 1/2

Thanks for your help.

Best Regards,
Sri

Like this post

gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pmFollowed by: 8 members; Thanked: 11 times

by gmattesttaker2 » Fri Jul 26, 2013 7:00 pm

Brent@GMATPrepNow wrote:
NYC493 wrote:Can someone tell me a quick way to decode a problem like this? If you start to solve each number in the sequence manually, you can see that it's alternating from positive to negative. But, how do you arrive at an approximate sum quickly?

"For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is given by (-1)^(k+1) x (1/2^k). If "T" is the sum of the first 10 terms in the sequence, then T is:

A. greater than 2
B. between 1 and 2
C. between 1/2 and 1
D. between 1/4 and 1/2
E. less than 1/2

Thanks!
Another option is to get a better idea of this sum, by finding a few terms:

T = 1/2 - 1/4 + 1/8 - 1/16 + . . .
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 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

Hello Brent,

Thanks a lot for the excellent explanation. I was clear till this point:

Of course, we can also see that T > 1/4

Is there a way we can arrive at this result similar to how you have concluded that T = A number less than 1/2

Thanks for your help.

Best Regards,
Sri

Like this post

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sat Jul 27, 2013 6:08 am

gmattesttaker2 wrote:I was clear till this point:

Of course, we can also see that T > 1/4
Is there a way we can arrive at this result similar to how you have concluded that T = A number less than 1/2

Thanks for your help.

Best Regards,
Sri

Hi Sri,

Once we get to this point... T = 1/4 + (a number less 1/4) ..., we can say that T = 1/4 + (some positive value).
If we add some positive value to 1/4, the sum (T) must be greater than 1/4

I hope that helps.

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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hemant_rajput: Master | Next Rank: 500 Posts; Posts: 447; Joined: Sun Apr 22, 2012 7:13 amFollowed by: 13 members; Thanked: 46 times; GMAT Score:700

by hemant_rajput » Sun Jul 28, 2013 8:54 am

here is my approach.

so first list down the term say 5 terms and see the pattern.

(1/2) , -(1/4),(1/8), -(1/16), (1/32)

so sum will be 1/2 - 1/4 + 1/8 - 1/16 +1/32...

add the number in pair of 2
so new series will be [1/2 - 1/4] + [1/8 - 1/16] +[ 1/32 - 1/64]...

1/4 + 1/16 + 1/64 ...
this is an infinite series and a Geometric progression too.

you can always use this simple formula to find the sum of a infinite GP series.

a/(1 - r), where a is the first term and r is the progressing factor.

a = 1/4

r = (1/4) / (1/16) = 1/4

so answer will be (1/4)/ [1 - (1/4)] = (1/4)/(3/4) = 1/3

which satisfy option D

I'm no expert, just trying to work on my skills. If I've made any mistakes please bear with me.

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