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Thread: 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 = $\displaystyle \frac{100}{2}(2a+(100-1)d)=10000$

    gives $\displaystyle 2a + 99d = 200$

    1st term = $\displaystyle a$

    2nd term = $\displaystyle a+d$

    5th term= $\displaystyle a+4d$

    Three terms of a geometric so exists $\displaystyle r$ such that

    $\displaystyle ar = a + d$ and $\displaystyle ar^2=a+4d$

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

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

    So $\displaystyle r=3$

    So $\displaystyle 3a = a+d$ and $\displaystyle 2a = d$

    Sub into equation you got $\displaystyle d + 99d = 100d = 200$

    $\displaystyle d=2$

    $\displaystyle a = 1$
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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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