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Thread: Formula for AP/GP?

  1. #1
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    Formula for AP/GP?

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    80.1/2+1/4+1/8+...+1/512=?
    A.0<S<1/2
    B.1/2<S<1
    C.1<S<3/2
    D. 3/2<S<2
    E. 2<S<5/2


    Don't know if this is a GP/AP problem. What is the formula for GP/AP. Also what is the difference between GP and AP?

  2. #2
    An Urch Guru Pundit Swami Sage
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    1/2+1/4+1/8+...+1/512=?....the terms int his sequence are in Geometric progression,
    first term a1 = 1/2
    r = a(n+1)/a(n) = 1/2
    number of terms (n) = 512/2 = 256

    Sum or n terms in GP is given by a1(1-r^n)/1-r
    = 1/2*[1-1/2^256]/1-1/2 = 1-1/2^256.

    Hence, IMO it is B.

  3. #3
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    Could you please elaborate or explain the details of your answer.

    Thanks

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    An Urch Guru Pundit Swami Sage Makumajon's Avatar
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    Hi, 1/2 + 1/4 + 1/8 +.... is a GP. If the series extends upto infinity, then the sum is 1. How? For infinite GP,
    Sum = a/(1-r); a=1st term & r= ratio of successive terms
    =1/2/(1-1/2)=1.

    Now the sequence terminates at 1/512. So, the sum must be less than 1. However, the sequence starts at 1/2 and all are positive terms. So the sum must be greater than 1/2. The answer must be B.

    NOTE: Although it's not required in this case, you might want to know the actual sum here. Since it's a finite GP, the formula is

    SUM =a*(1-r^n)/(1-r)
    =1/2*{1-(1/2)^9}/(1-1/2) [1/2+(1/2)^2+(1/2)^3+.....+(1/2)^9 => 9 terms, not 256 terms]
    =(1-1/2^9)

    Whether you use a*(1-r^n)/(1-r) or a*(r^n-1)/(r-1) does not matter. For the purity of of mathematics, we use the first one when r<1 and the second one when r>1. But all the same result. +/+ or -/-.

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    Trying to make mom and pop proud gmatprep2008's Avatar
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    To calculate how many number of terms:
    Since, Last Term = First Term * r ^(n-1)
    therefore:
    1/512 = 1/2 * (1/2)^(n-1)
    therefore n = 9

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    Now the sequence terminates at 1/512. So, the sum must be less than 1. However, the sequence starts at 1/2 and all are positive terms. So the sum must be greater than 1/2. The answer must be B.

    Very nice approach Makumajon

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    Within my grasp! ranjeet_1975's Avatar
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    In a finite geometric progression, S < 1

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    Quote Originally Posted by ranjeet_1975 View Post
    In a finite geometric progression, S < 1
    Ranjeet you forgot to mention..only if both the common ratio and first number is a fraction...

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    Yes, for example not in the case : 1 + 3+ 9+ 27+......

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