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Math Help - Another AP/GP question

  1. #1
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    Another AP/GP question

    The sum of the first 100 terms of an arithmetic progression is 10000; the first, second and fifth terms of this progression are three consecutive terms of a geometric progression. Find the first term a and the non-zero common difference, d, of the arithmetic progression. (Answer: a = 1, d = 2)

    I found:

    2a + 99d = 200

    After which, I'm stuck.


    Thanks in advance if you could help with this question!
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  2. #2
    Member Glaysher's Avatar
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    Sum of first 100 terms = \frac{100}{2}(2a+(100-1)d)=10000

    gives 2a + 99d = 200

    1st term = a

    2nd term = a+d

    5th term= a+4d

    Three terms of a geometric so exists r such that

    ar = a + d and ar^2=a+4d

    So  ar^2 = ar + 3d by subing in first equation into second

    Also ar^2 = ar + rd by multiplying both sides of first equation by r

    So r=3

    So 3a = a+d and 2a = d

    Sub into equation you got d + 99d = 100d = 200

    d=2

    a = 1
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  3. #3
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    Yay thanks glaysher now I get it!
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  4. #4
    Member Glaysher's Avatar
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    Quote Originally Posted by margaritas
    Yay thanks glaysher now I get it!
    No problem
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