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sum of terms in a sequence

This topic has 4 expert replies and 2 member replies
josh80 Senior | Next Rank: 100 Posts Default Avatar
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sum of terms in a sequence

Post Wed Dec 11, 2013 5:39 pm
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    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

    greater than 2
    between 1 and 2
    between 1/2 and 1
    between 1/4 and 1/2
    less than 1/4

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    Post Wed Dec 11, 2013 5:53 pm
    josh80 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 and 1
    D) between 1/4 and 1/2
    E) less than 1/4
    One way to solve this question is to apply the formula for what's known as a "geometric series," but I'm not really a fan of memorizing formulas. Another option is to get a better idea of this sum, by first 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 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, 1/4 < T < 1/2

    Answer: D

    Cheers,
    Brent

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    Amrabdelnaby Master | Next Rank: 500 Posts Default Avatar
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    Post Mon Nov 30, 2015 3:01 am
    Im little bit confused regarding this question.

    could you please explain how did you arrive to the first few terms?

    also this is a negative number. i thought the outcome will be once negative and once positive.

    the first term for example would be -1^(k+1)(1/2)^k = -1(2)(1/2) = -1.. so how come the first term is 1/4?

    please explain

    Brent@GMATPrepNow wrote:
    josh80 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 and 1
    D) between 1/4 and 1/2
    E) less than 1/4
    One way to solve this question is to apply the formula for what's known as a "geometric series," but I'm not really a fan of memorizing formulas. Another option is to get a better idea of this sum, by first 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 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, 1/4 < T < 1/2

    Answer: D

    Cheers,
    Brent

    Post Mon Nov 30, 2015 3:32 am
    Quote:
    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
    Calculate until you see the pattern.
    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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    Amrabdelnaby Master | Next Rank: 500 Posts Default Avatar
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    Post Mon Nov 30, 2015 5:07 am
    Ok I think I understand why i got it wrong now. thought that (k+1)(1/2^k) are both exponents of -1 while the fact is: -1^(k+1) x (1/2^k)

    hence for the first term it is -1^2 x (1/2) which is basically 1 x 1/2 = 1/2

    Thanks Guru Very Happy


    GMATGuruNY wrote:
    Quote:
    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
    Calculate until you see the pattern.
    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.

    Post Mon Nov 30, 2015 10:07 am
    Amrabdelnaby wrote:
    Im little bit confused regarding this question.

    could you please explain how did you arrive to the first few terms?

    also this is a negative number. i thought the outcome will be once negative and once positive.

    the first term for example would be -1^(k+1)(1/2)^k = -1(2)(1/2) = -1.. so how come the first term is 1/4?

    please explain

    kth term of a certain sequence = (-1)^(k+1) (1/2^k)

    term1 = (-1)^(1+1) (1/2^1)
    = [(-1)^2](1/2)
    = [1](1/2)
    = 1/2

    term2 = (-1)^(2+1) (1/2^2)
    = [(-1)^3](1/4)
    = [-1](1/4)
    = -1/4


    term3 = (-1)^(3+1) (1/2^3)
    = [(-1)^4](1/8)
    = [1](1/8)
    = 1/8

    Etc.

    So, we can write: T = 1/2 + (-1/4) + 1/8 + (-1/16) ....
    Or we can write: T = 1/2 - 1/4 + 1/8 - 1/16 ....

    Cheers,
    Brent

    _________________
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    Post Mon Nov 30, 2015 7:21 pm
    Hi All,

    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."

    You've probably correctly deduced 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

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